Answer:
b. {-9, 3}.
Step-by-step explanation:
x^2 + 6x - 27 = 0
We need 2 numbers whose product is -27 and whose sum is + 6.
These are + 9 and - 3. So the factors are:
(x - 3)(x + 9) = 0
(x - 3) = 0 or (x + 9) = 0
x = 3 or -9.
Answer: option b is the correct answer.
Step-by-step explanation:
The given quadratic equation is expressed as
x^2+6x-27=0
In order to solve the equation by factorization, the first step is to find two numbers such that their sum or difference is 6x and their product is
- 27x^2. The two numbers are 9x and - 3x. Therefore,
x^2 + 9x - 3x - 27 = 0
x(x + 9) - 3(x + 9) = 0
x - 3 = 0 or x + 9 = 0
x = 3 or x = - 9
Wagenlucht Ice Cream Company is always trying to create new flavors of ice cream. They are market testing three kinds to find out which one has the best chance of becoming popular. They give small samples of each to 20 people at a grocery store. 4 ice cream tasters preferred the Strawberry Cream, 12 preferred Choco- Nuts, and 4 loved the Orange Mint. Construct a Pareto chart to represent these preferences. Choose the vertical scale so that the relative frequencies are represented.
To construct a Pareto chart for the Wagenlucht Ice Cream Company, rank the flavors by preference, calculate relative frequencies, then draw a bar chart accordingly.
Explanation:The first step in constructing a Pareto chart is to order your categories (in this case, ice cream flavors) from largest to smallest frequency. Therefore, we will rank them as follows: Choco-Nuts (12), Strawberry Cream (4), and Orange Mint (4).
Then, calculate the relative frequencies - the number of people who preferred a particular flavor divided by the total number of people sampled. Choco-Nuts: 12/20 = 0.6, Strawberry Cream: 4/20 = 0.2, Orange Mint: 4/20 = 0.2.
Start a vertical bar chart with the flavors on the horizontal axis. Using the relative frequencies, draw proportional vertical bars for each: Choco-Nuts would be the tallest, then Strawberry Cream and Orange Mint, which are both the same size. This is your Pareto chart.
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The correct answer is option C. The relative frequency are as Choco-Nuts: 0.6, Strawberry Cream: 0.2, Orange Mint: 0.2.
To construct a Pareto chart representing the preferences for the Wagenlucht Ice Cream Company flavors, we need to follow these steps and choose an appropriate vertical scale. Here's the process:
1. Collect the data:
Strawberry Cream: 4 preferences
Choco-Nuts: 12 preferences
Orange Mint: 4 preferences
2. Calculate the total number of preferences:
[tex]\[ \text{Total preferences} = 4 + 12 + 4 = 20 \][/tex]
3. Calculate the relative frequencies:
Strawberry Cream: [tex]\(\frac{4}{20} = 0.2\)[/tex]
Choco-Nuts: [tex]\(\frac{12}{20} = 0.6\)[/tex]
Orange Mint: [tex]\(\frac{4}{20} = 0.2\)[/tex]
4. Order the categories in descending order of frequency:
Choco-Nuts: 60%
Strawberry Cream: 20%
Orange Mint: 20%
The complete question is:
Wagenlucht Ice Cream Company is always trying to create new flavors of ice cream. They are market testing three kinds to find out which one has the best chance of becoming popular. They give small samples of each to 20 people at a grocery store. 4 ice cream tasters preferred the Strawberry Cream, 12 preferred Choco- Nuts, and 4 loved the Orange Mint. Construct a Pareto chart to represent these preferences. Choose the vertical scale so that the relative frequencies are represented.
A. Choco-Nuts: 0.6, Strawberry Cream: 0.3, Orange Mint: 0.3.
B. Choco-Nuts: 0.4, Strawberry Cream: 0.4, Orange Mint: 0.1.
C. Choco-Nuts: 0.6, Strawberry Cream: 0.2, Orange Mint: 0.2.
D. Choco-Nuts: 0.6, Strawberry Cream: 0.4, Orange Mint: 0.2.
Powers of 10 make it very easy to write large and small numbers, but as a result it can also be easy to forget the large differences between different powers. We can compare any two numbers by dividing them. For example, we say that 12 is four times as large as 3 because 12÷3=4. Complete the sentences below comparing pairs of powers of 10.
Answer:let us compare the following pairs of power of 10,9×10^6 ÷3×10^12=3×10^-6
Step-by-step explanation:
Comparing pairs of power of 10 involve applying principle of indices.in what is known as the laws of indices
Law1 states that X^a ×X^b=X^(a+b) meaning that multiplication of indices results to addition of the indexes raise as exponenet of 10, similarly a division as in the answer above always lead to substraction of the indexes as seen in the example 9×10^6/3×10^12 will becomen9÷3×10^(6-12)=3×10^-6.
A body of constant mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k|v|, where k is a constant. Neglect changes in the gravitational force. a. Find the maximum height xm attained by the body and the time tm at which this maximum height is reached.
Answer:
tm = tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] }
Xm = Xₐ = (v₀m)/k - ({m²g}/k²) ㏑(1+{kv₀/mg})
Step-by-step explanation:
Note, I substituted maximum time tm = tₐ and maximum height Xm = Xₐ
We will use linear ordinary differential equation (ODE) to solve this question.
Remember that Force F = ma in 2nd Newton law, where m is mass and a is acceleration
Acceleration a is also the rate of change in velocity per time. i.e a=dv/dt
Therefore F = m(dv/dt) = m (v₂-v₁)/t
There are two forces involved in this situation which are F₁ and F₂, where F₁ is the gravitational force and F₂ is the air resistance force.
Then, F = F₁ + F₂ = m (v₂-v₁)/t
F₁ + F₂ = -mg-kv = m (v₂-v₁)/t
Divide through by m to get
-g-(kv/m) = (v₂-v₁)/t
Let (v₂-v₁)/t be v¹
Therefore, -g-(kv/m) = v¹
-g = v¹ + (k/m)v --------------------------------------------------(i)
Equation (i) is a inhomogenous linear ordinary differential equation (ODE)
Therefore let A(t) = k/m and B(t) = -g --------------------------------(ia)
b = ∫Adt
Since A = k/m, then
b = ∫(k/m)dt
The integral will give us b = kt/m------------------------------------(ii)
The integrating factor will be eᵇ = e ⁽k/m⁾
The general solution of velocity at any given time is
v(t) = e⁻⁽b⁾ [ c + ∫Beᵇdt ] --------------------------------------(iiI)
substitute the values of b, eᵇ, and B into equation (iii)
v(t) = e⁻⁽kt/m⁾ [ c + ∫₋g e⁽kt/m⁾dt ]
Integrating and cancelling the bracket, we get
v(t) = ce⁻⁽kt/m⁾ + (e⁻⁽kt/m⁾ ∫₋g e⁽kt/m⁾dt ])
v(t) = ce⁻⁽kt/m⁾ - e⁻⁽kt/m⁾ ∫g e⁽kt/m⁾dt ]
v(t) = ce⁻⁽kt/m⁾ -mg/k -------------------------------------------------------(iv)
Note that at initial velocity v₀, time t is 0, therefore v₀ = v(t)
v₀ = V(t) = V(0)
substitute t = 0 in equation (iv)
v₀ = ce⁻⁽k0/m⁾ -mg/k
v₀ = c(1) -mg/k = c - mg/k
Therefore c = v₀ + mg/k ------------------------------------------------(v)
Substitute equation (v) into (iv)
v(t) = [v₀ + mg/k] e⁻⁽kt/m⁾ - mg/k ----------------------------------------(vi)
Now at maximum height Xₐ, the time will be tₐ
Now change V(t) as V(tₐ) and equate it to 0 to get the maximum time tₐ.
v(t) = v(tₐ) = [v₀ + mg/k] e⁻⁽ktₐ/m⁾ - mg/k = 0
to find tₐ from the equation,
[v₀ + mg/k] e⁻⁽ktₐ/m⁾ = mg/k
e⁻⁽ktₐ/m⁾ = {mg/k] / [v₀ + mg/k]
-ktₐ/m = ㏑{ [mg/k] / [v₀ + mg/k] }
-ktₐ = m ㏑{ [mg/k] / [v₀ + mg/k] }
tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] }
Therefore tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] } ----------------------------------(A)
we can also write equ (A) as tₐ = m/k ㏑{ [mg/k] [v₀ + mg/k] } due to the negative sign coming together with the In sign.
Now to find the maximum height Xₐ, the equation must be written in terms of v and x.
This means dv/dt = v(dv/dx) ---------------------------------------(vii)
Remember equation (i) above -g = v¹ + (k/m)v
Given that dv/dt = v¹
and -g-(kv/m) = v¹
Therefore subt v¹ into equ (vii) above to get
-g-(kv/m) = v(dv/dx)
Divide through by v to get
[-g-(kv/m)] / v = dv / dx -----------------------------------------------(viii)
Expand the LEFT hand size more to get
[-g-(kv/m)] / v = - (k/m) / [1 - { mg/k) / (mg/k + v) } ] ---------------------(ix)
Now substitute equ (ix) in equ (viii)
- (k/m) / [1 - { mg/k) / (mg/k + v) } ] = dv / dx
Cross-multify the equation to get
- (k/m) dx = [1 - { mg/k) / (mg/k + v) } ] dv --------------------------------(x)
Remember that at maximum height, t = 0, then x = 0
t = tₐ and X = Xₐ
Then integrate the left and right side of equation (x) from v₀ to 0 and 0 to Xₐ respectively to get:
-v₀ + (mg/k) ㏑v₀ = - {k/m} Xₐ
Divide through by - {k/m} to get
Xₐ = -v₀ + (mg/k) ㏑v₀ / (- {k/m})
Xₐ = {m/k}v₀ - {m²g}/k² ㏑(1+{kv₀/mg})
Therefore Xₐ = (v₀m)/k - ({m²g}/k²) ㏑(1+{kv₀/mg}) ---------------------------(B)
The question is about an object projected upwards under gravity and a certain resistance. The equations of motion will be non-linear due to the nature of the resistance. Solving these equations metaphorically or numerically will yield the maximum height and time taken to reach that height.
Explanation:The subject matter here is mechanics which falls under Physics. Given that there is a body of constant mass m projected upwards with an initial velocity v0 and the medium being passed through provides a resistance of k|v|, the equations of motion under this resistance will be non-linear.
The question here pertains to the calculations related to an object moving upwards under a given resistance and gravity. To obtain the maximum height achieved by the body xm and the time taken to reach that tm, we employ the trick of non-dimensionalisation. First, we observe the units of all physical quantities and using this, we can introduce reduced physical quantities which are dimensionless.
Unfortunately, these non-linear equations don’t have a neat analytical solution, and methods of approximation or numerical techniques might be necessary to solve them for particular initial conditions.
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The circle graph shows Tommy Blox spent the money he earned last summer. If he spent $80 on entertainment, how much did Tommy earn altogether?
Answer:19+25+10+14+80 gives you 148
Step-by-step explanation:
Tommy earned approximately $68 altogether.
Given:
- Clothes: 19%
- Food: 25%
- Savings: 10%
- Other: 14%
To find out how much Tommy earned altogether, we need to detemine what percentage of his earnings $80 on entertainment represents.
First, we sum up these percentages to find out what portion of his earnings $80 represents:
Total percentage spent = Clothes + Food + Savings + Other
Total percentage spent = 19% + 25% + 10% + 14%
Total percentage spent = 68%
Now, we need to find out how much $80 represents as a percentage of his total earnings:
Percentage of earnings spent on entertainment = (Amount spent on entertainment / Total percentage spent) * 100%
Percentage of earnings spent on entertainment = (80 / 68) * 100%
Percentage of earnings spent on entertainment ≈ 117.65%
Now, to find out how much Tommy earned altogether, we need to determine the total amount represented by 100%, which is his total earnings. Since $80 represents approximately 117.65% of his earnings:
Total earnings = (Amount spent on entertainment / Percentage of earnings spent on entertainment) * 100%
Total earnings = (80 / 117.65%) * 100%
Total earnings ≈ $68
Therefore, Tommy earned approximately $68 altogether.
Accuracy refers to how closely the measured value of a quantity corresponds to its "true" value.
Precision expresses the degree of reproducibility or agreement between repeated measurements.
The more measurements you make and the better the precision, the smaller the error will be.
Terms
In Physics, accuracy and precision are terms used to discuss the validity of measurements. Accuracy refers to how close a measured value is to its true value, while precision discusses the consistency or reproducibility of measurements. Good precision can reduce errors but does not necessarily improve accuracy.
Explanation:In the field of Physics, accuracy and precision are used to discuss the reliability of measurements taken during experiments. Accuracy refers to how closely the measured value of a quantity corresponds to its 'true' value. For example, if we aim to measure a length of 10 meters, an accurate measurement would be as close to 10 meters as possible.
On the other hand, precision represents the degree of similarity, or reproducibility, between repeated measurements. If we measure the same length of 10 meters multiple times and get results like 9.9m, 10.1m, 10.0m, 9.9m, the measurements would be considered precise because they are closely clustered together, even if they are not necessarily 'accurate' (exact 10m).
It's important to understand that better precision does not always ensure better accuracy. The more measurements you make and the better the precision, the smaller the error will be, but accuracy requires measurements to be close to the actual value.
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F(x)=-1/4x-3, find the value of x for which f(x)=x
Answer:
x = -2.4
Step-by-step explanation:
f(x) = -1/4x -3
x = -1/4x -3 . . . . . the desired value of f(x)
5/4x = -3 . . . . . . . add 1/4x
x = -12/5 . . . . . . . multiply by 4/5, the inverse of 5/4
__
Check
-1/4(-2.4) -3 = 0.6 -3 = -2.4 = x . . . . answer checks OK
Max is a diver. He uses positive numbers to represent elevations above the water's surface and negative numbers to represent elevations below the water's surface. Max is standing on the springboard. He represent his location as 3 meters. He lets a ring drop to the bottom of the pool. He represents its location at the bottom of the pool as -4 meters. How many meters below the surface of the water is the ring? A. Less than 4 meters B. More than 4 meters C. Exactly 4 meters D. Exactly 3 meters Please help: :)
Write an expression that computes the average of the values 12 and 40, and assign it to the variable avg, which has already been defined.
Answer:
The average of 12 and 40 is 26
Step-by-step explanation:
The average of the values 12 and 40
The variable of average is avg
Average of numbers is sum of all number divide by number of numbers.
Expression:-
[tex]\text{Avg}=\dfrac{12+40}{2}[/tex]
Now simplify the average
[tex]\text{Avg}=\dfrac{52}{2}[/tex]
[tex]\text{Avg}=26[/tex]
Hence, the average of 12 and 40 is 26
The average of the values 12 and 40 is calculated by adding the two numbers together and dividing by 2. This computation can be assigned to a variable named 'avg' in a Python programming context.
Explanation:To compute the average of the values 12 and 40, you sum the two numbers and then divide by the count of numbers. In this case, there are two numbers, so the sum (12 + 40) is divided by 2. Thus, in the programming language Python, you could write this as:
avg = (12 + 40) / 2
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TRIANGLE ABC~EDC
What is the value of x?
Answer:
3.25
Step-by-step explanation:
36:24 = 2:3
6x-6 = (2/3)3x+7
x=3.25
Answer:
11=x
Step-by-step explanation:
AB AC
=
ED EC
36:24=6x−6:3x+7
108x+252=144x−144
396=36x
11=x
Find the perimeter of a triangle with sides measuring 3 centimeters, 4 centimeters and 5 centimeters.
a.
20 cm
c.
19 cm
b.
12 cm
d.
14 cm
Answer:
12
Step-by-step explanation:
P=a+b+c P=3+4+5 P=12
The perimeter of a triangle is 12 cm.
Option B is the correct answer.
What is a triangle?A triangle is a 2-D figure with three sides and three angles.
The sum of the angles is 180 degrees.
We can have an obtuse triangle, an acute triangle, or a right triangle.
We have,
There are three sides to a triangle.
So,
The sides are 3 cm, 4 cm, and 5 cm.
Now,
The perimeter of a triangle.
= Sum of all the sides
= 3 + 4 + 5
= 12 cm
Thus,
The perimeter of a triangle is 12 cm.
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Which is the graph of the function f(x) = Negative StartRoot x EndRoot
The graph of the function [tex]f(x)=-\sqrt{x}[/tex] is the first graph which is attached below.
Step-by-step explanation:
The function is [tex]f(x)=-\sqrt{x}[/tex]
To graph the function, we need to know the domain and range of the function.
The domain is found by substituting the values for x.
Thus, the domain is [tex]x\geq 0[/tex]
The range of the function is determined as [tex]y\leq 0[/tex]. Since, substituting the values of x we get the corresponding y-value which lies in the interval [tex](-\infty, 0][/tex].
The graph of the function [tex]f(x)=-\sqrt{x}[/tex] is the first graph which is attached below.
Apply the appropriate mathematical operation to solve this wheel and axle problem. Diameter of axle = 3.5" Axis of handle = 21" Weight lifted = 180 lb. Force, F = 15 a0 lb.
Answer:Force=30N
Step-by-step explanation:Torque of a force on the handle= Torque of weight on the axle.
Torque is the magnitude force that acts perpendicular.
3.5 ×180=21×Force
Force= 3.5×180 /21
Force=630/21
Force=30N
Answer:
6.74 lbs.
Step-by-step explanation:
Hope this helps.
Choose the correct solution graph for the inequality.
The correct answer is: Option number 4 (Last Option)
Step-by-step explanation:
Given inequality is:
-6x > 42
In order to solve the inequality,
Dividing both sides by 6
[tex]-\frac{6x}{6} > \frac{42}{6}\\-x > 7[/tex]
Multiplying by -1
[tex]x<7[/tex]
As the solution is x<7, this means that the number 7 will not be included in the solution and all numbers less than 7 will be a part of the solution.
The number which is not included in the solution is marked by a shallow circle on the number line.
Hence,
The correct answer is: Option number 4 (Last Option)
Keywords: Number line, inequality
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Given the pay rate and hours worked, determine the gross earnings. Make sure to include decimals and appropriate zeros. rate 6.50 hours 40
Final answer:
To determine the gross earnings for 40 hours worked at a pay rate of $6.50 per hour, multiply the pay rate by hours. The gross earnings would be $260.00.
Explanation:
To calculate the gross earnings given the pay rate and hours worked, we use a simple multiplication. However, there is an additional consideration mentioned in Exercise 3.1, which states that the employee should receive 1.5 times the hourly rate for hours worked above 40 hours. Therefore, the calculation involves two steps if the number of hours exceeds 40.
Calculation:
If hours ≤ 40, the formula is:In this particular case, the student only worked 40 hours at a pay rate of $6.50 per hour. Using the first formula, the gross earnings would be:
Gross Earnings = $6.50/hour × 40 hours = $260.00
A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: blue, green, yellow, or pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?
A. 6.
B. 8.
C. 16.
D. 24.
E. 32.
Answer:
Option C. 16
Step-by-step explanation:
Number of differents colors = 4
Number of differents sizes = 2
Case 1: 3 notepads of the same size and the same color:
If we have a package with the same size and the same color, the number of possible packages is:
N° packages = 4(colors)*2(sizes) = 8
Case 2: 3 notepads of the same size and different colors:
In this case, to calculate the number of possible permutations of packages without repetitions we need to use the following equation:
[tex] C_{n}^[p} = \frac{n!}{p!(n-p)!} [/tex]
where p: is the number of colors for each package = 3, and n: is the total number of colors = 4.
[tex] C_{4}^[3} = \frac{4!}{3!(4-3)!} = \frac{4*3*2*1}{3*2*1} = 4 [/tex]
This number calculated is for one size, if the have two different sizes the number of possible packages is:
N° packages = 4(colors)*2(sizes) = 8
Therefore, the total number of different possible packages is:
N° packages = case 1 + case 2 = 8 + 8 = 16
So, the correct answer is option C = 16.
I hope it helps you!
A company's annual profit, P, is given by P=−x2+195x−2175, where x is the price of the company's product in dollars. What is the company's annual profit if the price of their product is $32?
Answer: the company's annual profit if the price of their product is $32 is $3041
Step-by-step explanation:
A company's annual profit, P, is given by P = −x²+ 195x − 2175, where x is the price of the company's product in dollars.
To determine the company's annual profit if the price of their product is $32, we would substitute x = 32 into the given equation. It becomes
P = −32²+ 195 × 32 − 2175
P = −1024 + 6240 − 2175
P = $3041
Find four numbers that form a geometric progression such that the third term is greater than the first by 12 and the fourth is greater than the second by 36.
Answer:
5 , 4.5, 13.5 and 40.5
Step-by-step explanation:
Since the numbers are in geometric progression, their form is essentially:
a, ar, ar^2 and ar^3
Where a and r are first term and common ratio respectively.
From the information given in the catalog:
Third term is greater than the first by 12 while fourth is greater than second by 36.
Let’s now translate this to mathematics.
ar^2 - a = 12
ar^3 - ar = 36
From 1, a(r^2 - 1) = 12 and 2:
ar(r^2 - 1) = 36
From 2 again r[a(r^2 -1] = 36
The expression inside square bracket looks exactly like equation 1 and equals 12.
Hence, 12r = 36 and r = 3
Substituting this in equation 1,
a( 9 - 1) = 12
8a = 12
a = 12/8 = 1.5
Thus, the numbers are 1.5, (1.5 * 3) , (1.5 * 9), (1.5 * 27) = 1.5 , 4.5, 13.5 and 40.5
Final Answer:
The four numbers forming the geometric progression are 1.5, 4.5, 13.5, and 40.5.
Explanation:
Let's start by defining what a geometric progression (GP) is. A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Let's denote the four numbers in the GP as a, ar, ar², and ar³, where:
- a is the first term,
- r is the common ratio.
We've been given two conditions:
1. The third term is greater than the first by 12, which gives us the equation:
ar² = a + 12
2. The fourth term is greater than the second by 36, which leads us to:
ar³ = ar + 36
We need to solve this system of equations to find the values of a and r.
Starting with the first equation:
ar² = a + 12
We can subtract 'a' from each side to get:
ar - a = 12
Factor out 'a' from the left side:
a(r² - 1) = 12
Now notice that r² - 1 is a difference of squares and can be factored to (r + 1)(r - 1):
a(r + 1)(r - 1) = 12
This equation tells us that the product of 'a' and (r + 1)(r - 1) is 12. For now, let's keep this equation aside and look at the second condition.
Proceeding with the second equation:
ar = ar + 36
Subtract 'ar' from each side:
ar³ - ar = 36
Factor out 'ar':
ar(r² - 1) = 36
Again, we recognize a difference of squares in the parentheses, so we factor it:
ar(r + 1)(r - 1) = 36
This equation relates 'ar', and (r + 1)(r - 1), and tells us the product is 36.
Now, because we have a similar term in both equations, (r + 1)(r - 1), we can set the products equal to each other to find a relationship between 'a' and 'ar':
From the first equation, we have a(r + 1)(r - 1) = 12,
From the second equation, we have ar(r + 1)(r - 1) = 36.
Dividing the second equation by the first one gives us:
ar(r + 1)(r - 1) / a(r + 1)(r - 1) = 36 / 12
ar / a = 36 / 12
r = 3
Now that we have the value of 'r', let's substitute it back into either of the original equations to find 'a'. Let's use the first equation:
a(r² - 1) = 12
a(3² - 1) = 12
a(9 - 1) = 12
a(8) = 12
a = 12 / 8
a = 3 / 2
a = 1.5
Now we have both 'a' and 'r', which allows us to determine the four numbers in the GP:
The first number, a, is 1.5.
The second number, ar, is 1.5 * 3 = 4.5.
The third number, ar², is 4.5 * 3 = 13.5.
The fourth number, ar², is 13.5 * 3 = 40.5.
So, the four numbers forming the geometric progression are 1.5, 4.5, 13.5, and 40.5.
The number of ducks and pigs in a field totals 34. The total number of legs among them is 86. Assuming each duck has exactly two legs and each pig has exactly four legs, determine how many ducks and how many pigs are in the field. (For each answer, enter an exact number.)
Answer: the number of ducks in the field is 25
the number of pigs in the field is 9
Step-by-step explanation:
Let x represent the number of ducks in the field.
Let y represent the number of pigs in the field.
A duck has one head and a pig also has one head.
The number of ducks and pigs in a field totals 34. This means that
x + y = 34
The total number of legs among them is 86. Assuming each duck has exactly two legs and each pig has exactly four legs, it means that
2x + 4y = 86 - - - - - - - - - - -- - 1
Substituting x = 34 - y into equation 1, it becomes
2(34 - y) + 4y = 86
68 - 2y + 4y = 86
- 2y + 4y = 86 - 68
2y = 18
y = 18/2 = 9
Substituting y = 9 into x = 34 - y, it becomes
x = 34 - 9 = 25
To find the number of ducks and pigs in the field, we can set up a system of equations and solve them. Using the given information and the equations x + y = 34 and 2x + 4y = 86, we can find that there are 25 ducks and 9 pigs in the field.
Explanation:To solve this problem, we can use a system of equations. Let x represent the number of ducks and y represent the number of pigs. From the given information, we can set up two equations:
x + y = 34 (equation 1)
2x + 4y = 86 (equation 2)
Now, we can solve the system of equations. We can start by multiplying equation 1 by 2 to eliminate the x variable:
2(x + y) = 2(34)
2x + 2y = 68
Next, we can subtract equation 2 from this new equation:
(2x + 2y) - (2x + 4y) = 68 - 86
-2y = -18
Dividing both sides of the equation by -2 gives us:
y = 9
Substituting this value back into equation 1:
x + 9 = 34
x = 34 - 9
x = 25
Therefore, there are 25 ducks and 9 pigs in the field.
Steve starts his hike at an elevation of -261 feet below sea level what was the change in elevation from the start of his hike to the end and elevation is -108 feet below sea level
Answer:
153 feet
Step-by-step explanation:
The change in elevation is the difference between his ending elevation and his starting elevation:
-108 -(-261) = 153 . . . feet
Emil's backpack weighs six and three eights pounds. He removes a book that weighs three fourth pound. Then he removes a book that weighs one half pound .How much does Emil's back pack weigh now
Answer:
Emil's back pack weigh now [tex]5\frac{1}{8}\ pounds[/tex].
Step-by-step explanation:
Given:
Total Weight of backpack = [tex]6\frac{3}{8}\ pounds[/tex]
[tex]6\frac{3}{8}\ pounds[/tex] can be Rewritten as [tex]\frac{51}{8}\ pounds[/tex]
Weight of backpack = [tex]\frac{51}{8}\ pounds[/tex]
Weight of Book 1 = [tex]\frac{3}{4}\ pound[/tex]
Weight of Book 2 = [tex]\frac{1}{2}\ pound[/tex]
We need to find weight of back pack after removing books.
Solution:
Now we can say that;
weight of back pack after removing books can be calculated by Subtracting Weight of Book 1 and Weight of Book 2 from Total Weight of backpack.
framing in equation form we get;
weight of back pack after removing books = [tex]\frac{51}{8}-\frac{3}{4}-\frac{1}{2}[/tex]
Now to solve the equation we will first make the denominator common using LCM.
weight of back pack after removing books =[tex]\frac{51\times1}{8\times1}-\frac{3\times2}{4\times2}-\frac{1\times4}{2\times4}=\frac{51}{8}-\frac{6}{8}-\frac{4}{8}[/tex]
Now the denominators are common so we will solve the numerator.
weight of back pack after removing books = [tex]\frac{51-6-4}{8}=\frac{41}{8}\ pounds \ \ OR \ \ 5\frac{1}{8}\ pounds[/tex]
Hence Emil's back pack weigh now [tex]5\frac{1}{8}\ pounds[/tex].
In the theory of relativity, the mass of a particle with velocity v ism = m01 − v2/c2,where m0 is the mass of the particle at rest and c is the speed of light. What happens as v → c−?
A. m ? m0B. m ? ?C. m ? 0D. m ? ??
Answer:
as v tends to c( speed of light), the mass of the particle moves towards an infinite value
Step-by-step explanation:
The concept applied here is the theory of relativity.
what the theory entails is the measurements of events i.e things that happen, where and when they happen and to what large extend are events seperated in space and time. Albert Einstein was the first to published his findings on the theory of relativity.
When velocity of particle approaches to velocity of light . then, mass of particle approaches to infinite value.
Theory of relativity:In theory of relativity, the mass of a particle with velocity v is given as,
[tex]m=\frac{m_{0}}{\sqrt{1-\frac{v^{2} }{c^{2} } } }[/tex]
where [tex]m_{0}[/tex] is the mass of the particle at rest and c is the speed of light.
When velocity v tends to velocity of light c.
[tex]m=\frac{m_{0}}{\sqrt{1-\frac{c^{2} }{c^{2} } } }\\\\m=\frac{m_{0}}{\sqrt{1-1} } \\\\m=\frac{m_{0}}{ 0} =\infty[/tex]
Hence, when velocity of particle approaches to velocity of light . then, mass of particle approaches to infinite value.
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The ages of 3 brothers are represented by consecutive integers. If the oldest brothers age is decreased by twice the youngest brother age the result is -19. How old is the youngest brother
Answer:
The youngest brother's age is 21 years.
Step-by-step explanation:
Given:
The ages of 3 brothers are consecutive integers.
If the oldest brothers age is decreased by twice the youngest brother age the result is -19
To find the age of the youngest brother.
Solution:
Let the age of youngest broth be = [tex]x[/tex] years
The ages are consecutive integers.
So, age of the next older brother will be = [tex](x+1)[/tex] years
The age of the oldest brother will be = [tex](x+2)[/tex] years
The oldest brothers age is decreased by twice the youngest brother age.
The above statement can be represented as:
⇒ [tex](x+2)-2x[/tex]
Simplifying.
⇒ [tex]x-2x+2[/tex]
⇒[tex]-x+2[/tex]
The result for the above expression = -19.
So, we have:
[tex]-x+2=-19[/tex]
Subtracting both sides by 2.
[tex]-x+2-2=-19-2[/tex]
[tex]-x=-21[/tex]
Multiplying both sides by -1.
∴ [tex]x=21[/tex]
Thus, the youngest brother's age is 21 years
A Ferris wheel has a diameter of 42 feet. It rotates 3 times per minute. Approximately how far will a passenger travel during a 5-minute ride?a. 132 feet
b. 659 feet
c. 1,978 feet
d. 3,956 feet
Step-by-step explanation:
Diameter, D = 42 feet
Circumference = πD = π x 42 = 131.95 feet
Number of rotations per minute = 3
Total time = 5 minutes
Total rotations = 5 x 3 = 15
Distance traveled per rotation = 131.95 feet
Distance traveled in 15 rotations = 15 x 131.95 = 1978 feet
Option C is the correct answer.
Do Now 60: What are the formulas to find area for a square, triangle, rectangle, parallelogram, trapezoid, circle, ellipse and equilaterial triangle?
Answer:
Area of a square = Length × Length
Area of a triangle = 1/2 base × height
Area of a rectangle = Length × breadth
Area of a parallelogram = base × height
Area of a trapezoid = 1/2 × sum of parallel sides × height
Area of circle = π × square of the radius
Area of ellipse = π × product of major and minor radii
Area of equilateral triangle = 1/2 base × height
Step-by-step explanation:
The area of a square is calculated by multiplying the length by itself.
The area of a triangle is calculated by multiplying half the base of the triangle by its height
The area of a rectangle is found by multiplying the length of the rectangle by its breadth
The area of a parallelogram is calculated by multiplying the base of the parallelogram by its height
The area of a trapezoid is found by multiplying half the sum of the two parallel sides by its height
The area of a circle is calculated by multiplying pi by the square of the radius of the circle
The area of an ellipse is found by multiplying pi by the product of the major and minor radii of the ellipse
The area of an equilateral triangle is calculated by multiplying half the base of the triangle by its height. The height is calculated using Pythagoras theorem
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. Is this statement true or false?A) True B) False
Answer:
A) True
Step-by-step explanation:
A basic variable is a variable that corresponds to a pivot column. A variable that does not corresponds to a pivot column is known as a free variable. It is required to row reduce the augmented matrix into echelon form so as to determine which of the variables are free and which of them are basic.
The statement is true. A basic variable in a linear system corresponds to a pivot column in the coefficient matrix. Basic variables have the leading ones in the matrix, while the rest are free variables.
Explanation:The statement 'A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.' is true. In a linear system, the basic variables are indeed those that correspond to the pivot column in the coefficient matrix. For example, if we have a linear system represented by a coefficient matrix, the basic variables would be the variables associated with the columns that have a leading one (also known as the pivot). The rest of the variables are known as free variables.
Consider a matrix with three columns representing three variables: x, y, and z. If x and y have leading ones and z does not, x and y would be the basic variables, whilst z would be a free variable. Therefore, understanding the relationship between the basic variables and the pivot column in a system's coefficient matrix is crucial in solving linear systems.
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Describe the set of points whose distance from the y-axis equals the distance from the xz-plane. a cone opening along the x-axis a cone opening along the y-axis a cone opening along the z-axis a cylinder opening along the x-axis a cylinder opening along the y-axis a cylinder opening along the z-axis Find an equation for the set of points.
The correct option is option D , i.e., a cone opening along y - axis.
It is given that a set of points is there whose distance from the y-axis equals the distance from the x-z plane.
We have to find out which of the given options describe these set of points.
What is a cone ?
A cone is a series of tapering circular plates which stack over each other from a flat base to a point.
As per the question ;
Distance from y-axis = Distance from the x-z plane
y = [tex]\sqrt{x^{2} + z^{2} }[/tex]
y² = x² + z²
So , this is a cone that has its vertex at the origin and opens along y-axis.
Thus , the correct option is option D , i.e., a cone opening along y - axis.
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What is the median value of the set R, if for every term in the set, [tex]R_n = R_{n-1} + 3[/tex]? (1) The first term of set R is 15. (2) The mean of set R is 36.
Answer:
36
Step-by-step explanation:
Given that
[tex]R_n = R_{n+1} +3[/tex] is given
First term is 15
This is an arithmetic series with a =15 and d =3
If n is the number of terms, then we have
Sum of n terms = 36 xn = 36n
But as per arithmetic progression rule
[tex]S_n = \frac{n}{2} [2a+(n-1)d]\\= \frac{n}{2} [30+(n-1)3]=36n[/tex]
[tex]72 = 30+3n-3\\n-=15[/tex]
When there are n terms we have middle term is the 8th term
Hence median is 8th term
=[tex]a_8 = 15+7(3) \\=36[/tex]
A recent report stated "Based on a sample of 170 truck drivers, there is evidence to indicate that, on average, independent truck drivers earn more than company-hired truck drivers." Does this statement describe descriptive or inferential statistics?
Answer:
The given statement describe inferential statistics.
Step-by-step explanation:
Descriptive Statistic:
It helps us to summarize a given data set.It could describe the entire population or a sample from the population.There are two types of descriptive measures: measures of central tendency and measures of variabilityCentral Tendency: Mean, mode, MedianMeasure of Viability: Standard Deviation, Variance, Range, Interquartile rangeInferential Statistic:
It s the process of estimating population parameter with the help of a sample from the population.A random sample from the population is used to describe the population with the help of sample statistic.Given Scenario:
"Based on a sample of 170 truck drivers, there is evidence to indicate that, on average, independent truck drivers earn more than company-hired truck drivers."
Thus, this is an example of a inferential statistics as a sample was used to estimate the population.
Here,
Sample:
Sample of 170 truck drivers
Population:
All truck drivers.
With the help of a sample, we approximated the population, thus, this statement describe inferential statistics.
The statement is an example of inferential statistics, as it makes a general conclusion about a population (all truck drivers) based on a sample.
The statement "Based on a sample of 170 truck drivers, there is evidence to indicate that, on average, independent truck drivers earn more than company-hired truck drivers" describes the use of inferential statistics. This type of statistics is used when analysts want to make predictions or inferences about a population based on the data collected from a sample. In contrast, descriptive statistics are used simply to describe what the data show, such as calculating averages, medians, ranges, and so on. Since the statement indicates a broader conclusion about the earnings of independent versus company-hired truck drivers in general, based on a sample, it utilizes inferential statistics.
A store is mixing up two types of nuts, peanuts and cashews into a 50 lb barrel. peanuts sell for $4 a pound and cashews sell for $7 a pound. If the store wants to sell the mix for $5.75 a pound, how many pounds of each nut should be put into the mix?
Answer:
Cashew 12.5lb
Peanuts 37.5lb
Step-by-step explanation:
Let the number of pounds of cashewnuts and peanuts be c and p respectively.
Firstly, the total mass of the nuts is 50.
This means:
c + p = 50
Now let’s work with the money
4p + 7c = 4.75(50)
From the first equation, let c = 50 - p
Substitute this into the second equation.
4p + 7(50 - p) = 237.5
4p + 350 - 7p = 237.5
3p = 112.5
P = 112.5/3 = 37.5lb
For Cashew c = 50 - p = 50 - 37.5 = 12.5lb
1.44 Make-up exam: In a class of 28 students, 27 of them took an exam in class and 1 student took a make-up exam the following day. The professor graded the first batch of 27 exams and found an average score of 79 points with a standard deviation of 6.5 points. The student who took the make-up the following day scored 63 points on the exam.
a) Does the new student's score increase or decrease the average?
Decreases
Increases
b) The new average is: (round to two decimal places)
c) Does the new student's score increase or decrease the standard deviation of the scores?
Decreases
Increases
Answer:
a) Decrease
b) New mean = 78.43
c) Decrease
Step-by-step explanation:
We are given the following in the question:
Total number of students in class = 28
Average of 27 students = 79
Standard Deviation of 27 students = 6.5
New student's score = 63
a) The new student's score will decrease the average.
b) New mean
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean = \dfrac{\displaystyle\sum x_i}{27} = 79\\\\\sum x_i = 27\times 79 = 2133[/tex]
New mean =
[tex]\text{ New mean} =\dfrac{ \displaystyle\sum x_i +63}{28}\\\\ =\dfrac{2133+63}{28}= \dfrac{2196}{28} = 78.43[/tex]
Thus, the new mean is 78.43
c) Since the new mean decreases, standard deviation for new scores will decrease.
This is because the new value is within the usual values i.e. within two standard deviations of the mean. So, this wont cause a lot of variation as this value will be closer to already available data values. Also number of observations (n) in the denominator is increasing. Based on both these points we can conclude that standard deviation will decrease
Formula for Standard Deviation:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.