Answer:
1 is the positive number for which the sum of it and its reciprocal is the smallest.
Step-by-step explanation:
Let x be the positive number.
Then, the sum of number and its reciprocal is given by:
[tex]V(x) = x + \dfrac{1}{x}[/tex]
First, we differentiate V(x) with respect to x, to get,
[tex]\frac{d(V(x))}{dx} = \frac{d(x+\frac{1}{x})}{dx} = 1-\dfrac{1}{x^2}[/tex]
Equating the first derivative to zero, we get,
[tex]\frac{d(V(x))}{dx} = 0\\\\1-\dfrac{1}{x^2}= 0[/tex]
Solving, we get,
[tex]x^2 = 1\\x= \pm 1[/tex]
Since x is a positive number x = 1.
Again differentiation V(x), with respect to x, we get,
[tex]\frac{d^2(V(x))}{dx^2} = \dfrac{2}{x^3}[/tex]
At x = 1
[tex]\frac{d^2(V(x))}{dx^2} > 0[/tex]
Thus, by double derivative test minima occurs for V(x) at x = 1.
Thus, smallest possible sum of a number and its reciprocal is
[tex]V(1) = 1 + \dfrac{1}{1} = 2[/tex]
Thus, 1 is the positive number for which the sum of it and its reciprocal is the smallest.
A homeowner wants to increase the size of a rectangular deck that now measures 14 feet by 22 feet. The building code allows for a deck to have a maximum area of 800 square feet. If the length and width are increased by the same number of feet, find the maximum number of whole feet each dimension can be increased.
Answer:
10.6 feet
Step-by-step explanation:
Length = 22ft
width= 14ft
Maximum area = 800 sq. ft
Let X be increase in the number of feet for the length and width.
The new length = (22 + x) ft
New width= (14 + x) ft
Area = (22+x)(14+x) ≤ 800
308 + 36x + x^2 ≤ 800
x^2 + 36x + 308 - 800 ≤ 0
x^2 + 36x - 492 ≤ 0
Solve using quadratic equation
x = (-b +/- √b^2 - 4ac) / 2a
a= 1, b = 36, c= 492
x = (-36 +/- √36^2 - 4*1*-492)/ 2*1
= (-36 +/- √1396 + 1968) / 2
= (-36 +/- √3264) / 2
= (-36 +/- 57.13) / 2
x = (-36 + 57.13)/2 or (-36 - 57.13)/2
x = 21.13/2 or -93.13/2
x = 10.6 or -31.0
x = 10.6 ft
The length and width must increase by 10.6 ft each
The number of whole feet by which the length and width of the deck can be increased would be calculated by setting up and solving an equation taking into account the original dimensions and maximum allowed area for the deck.
Explanation:The question is asking to find by how many whole feet the length and width of the rectangular deck can be increased, given a maximum area limit set by the building code. Let's suppose x is the number of feet by which both the length and width are to be increased. Given that the original area of the deck is 14 feet by 22 feet (which gives an area of 308 square feet), the new dimensions would be (14 + x) feet by (22 + x) feet.
According to the building code, the maximum area allowed for the deck is 800 square feet. We can write and solve the following equation to find x: (14 + x) * (22 + x) = 800.
Solving this equation would yield the maximum integral number of feet each dimension can be increased while still not exceeding the allowed 800 square feet.
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A hockey player is offered two options for a contract: either a base salary of 50,000 and 1000 per goal, or a base salary of 40,000 and 1500 per hoal. How may goals must he score in order to make the same money as the other contract?
The hockey player must score 20 goals in order to make the same money as the first contract
Solution:
Given that, hockey player is offered two options for a contract
Let "x" be the number of goals made
Option 1 :A base salary of 50,000 and 1000 per goal
Therefore, money earned is given as:
Money earned = 50000 + 1000(number of goals)
Money earned = 50000 + 1000x ------- eqn 1
Option 2:Base salary of 40,000 and 1500 per goal
Therefore, money earned is given as:
Money earned = 40000 + 1500(number of goals)
Money earned = 40000 + 1500x --------- eqn 2
In order to make the same money as the other contract, eqn 1 must be equal to eqn 2
50000 + 1000x = 40000 + 1500x
1500x - 1000x = 50000 - 40000
500x = 10000
x = 20
Thus he must score 20 goals in order to make the same money as the first contract
I do not understand this question please help picture attached Which graph represents the function?
f(x)=x+2−−−−√3
the answer is (-2,0)
Find the root(s) of f (x) = (x + 5)3(x - 9)2(x + 1). -5 with multiplicity 3 5 with multiplicity 3 -9
Answer:
-5, multiplicity 3; +9, multiplicity 2; -1
Step-by-step explanation:
The roots of f(x) are those values of x that make the factors be zero. For a factor of x-a, the root is x=a, because a-a=0. If the factor appears n times, then the root has multiplicity n.
f(x) = (x+5)^3(x-9)^2(x+1) has roots ...
-5 with multiplicity 3+9 with multiplicity 2-1Answer:
a. -5 with multiplicity 3
d. 9 with multiplicity 2
f. -1 with multiplicity 1
Step-by-step explanation:
edge
Water flows out of a hose at a constant rate after 2 1/3 minutes 9 4/5 gallons of water had come out of the hose at what rate in gallons per minute is the water flowing out of the hose
Answer:
4 1/5 gallons per minute
Step-by-step explanation:
Divide gallons by minutes.
(9 4/5 gal)/(2 1/3 min) = (49/5 gal)/(7/3 min) = (49/5)(3/7) gal/min
= 21/5 gal/min = 4 1/5 gal/min
A countrys population in 1995 was 173 million in 1997 it was 178 million estamate the population in 2005 using the expontial growth formula round your awser to the nearest million
Answer: the population in 2005 is
199393207
Step-by-step explanation:
The formula for exponential growth which is expressed as
A = P(1 + r/n)^ nt
Where
A represents the population after t years.
n represents the periodic interval at which growth is recorded.
t represents the number of years.
P represents the initial population.
r represents rate of growth.
From the information given,
P = 173 million
A = 178 million
t = 1997 - 1995 = 2 years
n = 1
Therefore
178 × 10^6 = 173 × 10^6(1 + r/1)^2 × 1
178 × 10^6/173 × 10^6 = (1 + r)^2
1.0289 = (1 + r)^2
Taking square root of both sides, it becomes
√1.0289 = √(1 + r)^2
1 + r = 1.0143
r = 1.0143 - 1 = 0.0143
Therefore, in 2005,
t = 2005 - 1995 = 10
A = 173 × 10^6(1 + 0.0143)^10
A = 173 × 10^6(1.0143)^10
A = 199393207
A Ferris wheel 50ft in diameter makes one revolution every 40sec. If the center of the wheel is 30ft above the ground, how long after reaching the low point is a rider 50ft above the ground?
Answer:
The rider is 15.91 seconds 50 ft above the ground after reaching the low point
Step-by-step explanation:
We can evaluate the angle α
using trigonometry applied to the orange small triangle with height 50-30 = 20 ft and hypotenuse equal to the radius r = 25ft
Now
[tex]20 = \frac{25}{sin(\alpha)}[/tex]
[tex]\alpha = arcsin{\frac{20}{25}[/tex]
[tex]\alpha = 53.13^{\circ}[/tex]
So 50 ft of height corresponds to the total angle:
[tex]90^{\circ} =53.13^{\circ} = 143.13 6^{\circ}[/tex] = 2.498 radians
Now the angular velocity
[tex]\omega = \frac{2\pi}{T}[/tex]
[tex]\omega = \frac{2 \pi}{40}[/tex]
[tex]\omega =0.157rad/s[/tex]
To describe 2.498 rad it will take:
[tex]t = \frac{2.498}{0.157}[/tex]
t = 15.91 s
The rider reaches 50 feet above the ground approximately 4.09 seconds after reaching the lowest point.
To solve the problem of determining how long after reaching the low point a rider on the Ferris wheel is 50 feet above the ground, we can follow these steps:
Step 1: Define the parameters
Diameter of the Ferris wheel: 50 feet.
Radius of the Ferris wheel: [tex]\( r = \frac{50}{2} = 25 \)[/tex] feet.
Center of the wheel height: 30 feet above the ground.
Rotation period: 40 seconds per revolution.
Step 2: Establish the equation for the height of a rider
The vertical position ( h(t) ) of a rider at time ( t ) seconds can be modeled using the cosine function (assuming the lowest point at ( t = 0 ):
[tex]\[ h(t) = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]
Step 3: Set the height equation to 50 feet
Since we want the height h(t) to be 50 feet:
[tex]\[ 50 = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]
Step 4: Solve for the cosine term
[tex]\[ 50 = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]
[tex]\[ 20 = 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]
[tex]\[ \cos\left(\frac{2\pi t}{40}\right) = \frac{20}{25} \][/tex]
[tex]\[ \cos\left(\frac{2\pi t}{40}\right) = 0.8 \][/tex]
Step 5: Determine the angle
[tex]\[ \frac{2\pi t}{40} = \cos^{-1}(0.8) \][/tex]
[tex]\[ \frac{2\pi t}{40} = 0.6435 \][/tex]
[tex]\[ t = \frac{0.6435 \times 40}{2\pi} \][/tex]
[tex]\[ t \approx 4.09 \text{ seconds} \][/tex]
Thus, the rider reaches 50 feet above the ground approximately 4.09 seconds after reaching the lowest point.
If the probability is 0.1 that a person will make a mistake on his or her state income tax return, find the probability that
(a) four totally unrelated persons each make a mistake;
(b) Mr. Jones and Ms. Clark both make mistakes, and Mr. Roberts and Ms. Williams do not make a mistake.
Assume the events are independent.
Answer: a) 0.0001, b) 0.0081.
Step-by-step explanation:
Since we have given that
Probability that a person will make a mistake on their state income tax return = 0.1
Probability that a person will not make any mistake = [tex]1-0.1=0.9[/tex]
Since the events are independent.
So, (a) four totally unrelated persons each make a mistake;
Our probability becomes,
[tex]P(A\cap B\cap C\cap D)=P(A)\times P(B)\times P(C)\times P(D)\\\\=0.1\times 0.1\times 0.1\times 0.1\\\\=0.0001[/tex]
(b) Mr. Jones and Ms. Clark both make mistakes, and Mr. Roberts and Ms. Williams do not make a mistake.
So, our probability becomes,
[tex]P(A\cap B\cap C'\cap D')=P(A)\times P(B)\times P(C')\times P(D')\\\\=0.1\times 0.1\times 0.9\times 0.9\\\\=0.0081[/tex]
Hence, a) 0.0001, b) 0.0081.
Final answer:
The probability that four unrelated persons each make a mistake on their tax returns is 0.0001. The probability that Mr. Jones and Ms. Clark both make mistakes, while Mr. Roberts and Ms. Williams do not, is 0.0081, calculated by multiplying the individual probabilities of each event.
Explanation:
The probability of independent events occurring simultaneously is calculated by multiplying the probabilities of each individual event. For part (a), we need to find the probability that four totally unrelated persons each make a mistake on their tax returns. Since the probability of one person making a mistake is 0.1, and the events are independent, we multiply this probability four times.
Probability (four unrelated persons each make a mistake) = 0.1x0.1x0.1x0.1 = 0.0001
For part (b), we are looking for the probability that Mr. Jones and Ms. Clark both make mistakes, while Mr. Roberts and Ms. Williams do not make mistakes. The probability of making a mistake is 0.1 and not making a mistake is 0.9 (which is 1 - 0.1).
Probability (Jones and Clark make mistakes, Roberts and Williams do not) = 0.1x0.1x0.9x0.9 = 0.0081
An open box is to be constructed from a square piece of sheet metal by removing a square of side 2 feet from each corner and turning up the edges. If the box is to hold 32 cubic feet, what should be the dimensions of the sheet metal?
Answer:
length 8 feet and width 8 feet
Step-by-step explanation:
Lets x be the length and width of the sheet
An open box is to be constructed from a square piece of sheet metal by removing a square of side 2 feet from each corner
so height is 2 feet
length of the box is x-4 feet
width of the box is also x-4 feet
Volume of the box is length times width times height
[tex]V= (x-4)(x-4)2[/tex]
[tex]32=2 (x-4)^2[/tex]
divide both sides by 2
[tex]16=(x-4)^2[/tex]
take square root on both sides
[tex]4=x-4[/tex]
Add 4 on both sides
[tex]x=8[/tex]
So dimension of the sheet is
length 8 feet and width 8 feet
Point N lies on the side AB in △ABC. Points K and M are midpoints of segments
BC
and
AN
respectively. Segments
CN
and
MK
intersect at point T. It is known that ∠CTK≅∠TMN and AB=7. Find the length of segment
CN
.
Answer:
CN = 7
Step-by-step explanation:
In the attached figure, we have drawn line CD parallel to AB with D a point on line MK. We know ΔMNT ~ ΔDCT by AA similarity, and because of the given angle congruence, both are isosceles with CD = CT. Likewise, we know ΔCDK is congruent to ΔBMK by AAS congruence, since BK = CK (given).
Then CD = BM (CPCTC). Drawing line NE creates isosceles ΔNEC ~ ΔTDC and makes CE = AB. Because ΔNEC is isosceles, CN = CE = AB = 7.
The length of segment CN is 7.
_____
If you assume CN is constant, regardless of the location of point N (which it is), then you can locate point N at B. That also collocates points T and K and makes ΔBMK both isosceles and similar to ΔBAC. Then CN=AB=7.
And artificial lake is in the shape of a rectangle and has an area of 9/20 square mi.² the width of the lake is 1/5 the length of the lake what are the dimensions of the lake
The dimensions of lake are length 1.5 miles and width 0.3 miles
Solution:
Given that,
Artificial lake is in the shape of a rectangle
Let the length of lake be "a"
The width of the lake is 1/5 the length of the lake
[tex]width = \frac{length}{5}\\\\width = \frac{a}{5}[/tex]
The area of lake is 9/20 square miles
The area of rectangle is given by formula:
[tex]Area = length \times width[/tex]
Substituting the values we get,
[tex]\frac{9}{20} = a \times \frac{a}{5}\\\\a^2 = \frac{9}{20} \times 5\\\\a^2 = \frac{9}{4}\\\\\text{Taking square root on both sides }\\\\a = \frac{3}{2} = 1.5[/tex]
Thus, we get
[tex]length = a = 1.5 \text{ miles }\\\\width = \frac{a}{5} = \frac{1.5}{5} = 0.3 \text{ miles }[/tex]
Thus the dimensions of lake are length 1.5 miles and width 0.3 miles
The Poe family bought a house for $240,000. If the value of the house increases at a rate of 4% per year, about how much will the house be worth in 20 years?
Answer:
about $525,900
Step-by-step explanation:
Each year, the value is multiplied by (1 +4%) = 1.04. After 20 years, it will have been multiplied by that value 20 times. That multiplier is 1.04^20 ≈ 2.19112314.
The value of the house in 20 years will be about ...
$240,000×2.19112314 ≈ $525,900 . . . . . rounded to hundreds
Answer: it would be worth $52587 in 20 years.
Step-by-step explanation:
If the value of the house increases at a rate of 4% per year, then the rate is exponential. We would apply the formula for exponential growth which is expressed as
A = P(1 + r/n)^ nt
Where
A represents the value of the house after t years.
n represents the period of increase.
t represents the number of years.
P represents the initial value of the house.
r represents rate of increase.
From the information given,
P = $24000
r = 4% = 4/100 = 0.04
n = 1 year
t = 20 years
Therefore
A = 24000(1 + 0.04/1)^1 × 20
A = 24000(1.04)^20
A = $52587
Perform the following facility capacity problem. In planning the opening of your restaurant, you estimate that the restaurant's total area should allow each customer 20 square feet of dining space. Of the planned space, you hope to utilize one-third of the total space for the kitchen and storage. Seating capacity = 90 Area of dining room = a0 sq. ft. Area of kitchen and storage (to the nearest sq. ft.) = a1 sq. ft.
Answer:
Step-by-step explanation:
Total area = Dining space + Kitchen and Storage space
Dining space = 90 multiplied by 20 = 1800 square feet
Kitchen and Storage space = 1/3 of Total area
Total area = 1800 + 1/3 of Total area
(1-1/3) Total area = 1800
2/3 Total area = 1800
Total area = 3*1800/2 = 2700
Kitchen and Storage space = 2700*1/3 = 900 square feet
A new drug on the market is known to cure 24% of patients with colon cancer. If a group of 15 patients is randomly selected, what is the probability of observing, at most, two patients who will be cured of colon cancer? 15 choose 2(0.24)2(0.76)13 15 choose 0(0.76)15 + 15 choose 1(0.24)1(0.76)14 + 15 choose 2(0.24)2(0.76)13 1 − 15 choose 2(0.24)2(0.76)13 15 choose 0 (0.76)15 1 − 15 choose 0 (0.76)15
Answer:
P = ₁₅C₀ (0.76)¹⁵ + ₁₅C₁ (0.24)¹ (0.76)¹⁴ + ₁₅C₂ (0.24)² (0.76)¹³
Step-by-step explanation:
Binomial probability:
P = nCr pʳ qⁿ⁻ʳ
where n is the number of trials,
r is the number of successes,
p is the probability of success,
and q is the probability of failure (1−p).
Here, n = 15, p = 0.24, and q = 0.76.
We want to find the probability when r is at most 2, which means r = 0, r = 1, and r = 2.
P = ₁₅C₀ (0.24)⁰ (0.76)¹⁵⁻⁰ + ₁₅C₁ (0.24)¹ (0.76)¹⁵⁻¹ + ₁₅C₂ (0.24)² (0.76)¹⁵⁻²
P = ₁₅C₀ (0.76)¹⁵ + ₁₅C₁ (0.24)¹ (0.76)¹⁴ + ₁₅C₂ (0.24)² (0.76)¹³
The correct answer is When P = ₁₅C₀ (0.76)¹⁵ + ₁₅C₁ (0.24)¹ (0.76)¹⁴ + ₁₅C₂ (0.24)² (0.76)¹³
The first step is Binomial probability:P = nCr pʳ qⁿ⁻ʳ
Also, that where n is the number of trials,After that r is the number of successes,p is the probability of success, and also that q is the probability of failure (1−p).So that Here, n = 15, p = 0.24, and q = 0.76.
The second step is We want to find the probability when r is at most 2, which were means that the r = 0, r = 1, and r = 2.When P = ₁₅C₀ (0.24)⁰ (0.76)¹⁵⁻⁰ + ₁₅C₁ (0.24)¹ (0.76)¹⁵⁻¹ + ₁₅C₂ (0.24)² (0.76)¹⁵⁻²After that P = ₁₅C₀ (0.76)¹⁵ + ₁₅C₁ (0.24)¹ (0.76)¹⁴ + ₁₅C₂ (0.24)² (0.76)¹³Learn more information:
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plz, help ASAP!!!!!!!!!!
the options are
reflection across the x-axis or y-axis
and then a translation 4 units down, up, right left
Answer:
Reflection over the x-axis and translation 4 units left
Step-by-step explanation:
Carolina is twice as old as Raul. Ginny is the oldest and is three times Raul's age plus four. Their ages add up to be 52. How old will Ginny be on her next birthday?
Answer:
29
Step-by-step explanation:
Let c, r, g stand for the current ages of Carolina, Raul, and Ginny. Then we have the relations ...
c = 2r
g = 3r +4
c + r + g = 52
Substituting for c and g, we get ...
2r + r + (3r +4) = 52
6r = 48 . . . . . . . . . . . . . subtract 4
r = 8 . . . . . . . . . . . . . . . divide by 8
g = 3(8) +4 = 28 . . . . . .Ginny is presently 28
On Ginny's next birthday, she will be 29.
Answer: Ginny would be 29 years old on her next birthday.
Step-by-step explanation:
Let c represent Carolina's current age.
Let r represent Raul's current age.
Let g represent Ginny's current age.
Carolina is twice as old as Raul. It means that
c = 2r
Ginny is the oldest and is three times Raul's age plus four. It means that
g = 3r + 4
Their ages add up to be 52. It means that
c + r + g = 52 - - - - - - - - - - - - 1
Substituting
c = 2r and g = 3r + 4 into equation 1, it becomes
2r + r + 3r + 4 = 52
6r + 4 = 52
6r = 52 - 4 = 48
r = 48/6 = 8
c = 2r = 2 × 8
c = 16
g = 3r + 4 = 3 × 8 + 4
g = 24 + 4 = 28
On Ginny's next birthday, she would be 28 + 1 = 29 years old
PLZ HELP ME ASAP!!!!!!
WHAT I CHOOSE IS IT THE RIGHT ANSWER????
Answer:
YES!!!
Step-by-step explanation:
Answer:
Your answer is going to be C
Step-by-step explanation:
So, when it is on the top, your X will always be first and you can see it is not in the negatives. When it is on the bottom Y is first and you can see it is in the negatives.
When playing the 10 number game, we can use 1, 2, 3, 4, 5, 6, 7, 8, 9, 10(each number only one time), addition, multiplication, and parentheses. What is the largest number that we can create the 10-number game
Hailey has five times as many stuffed animals as she does dolls her brother has 18 more video games and Hailey has stuffed animals if Hailey has $15 how many video games does her brother have
Answer:
Her brother has 33 video games
Step-by-step explanation:
You take 15 plus 18 because hailey has 15 stuffed animals and her brother has 18 more video games then she has animals so you add 15+ 18 to get your final answer of 33 video games
The number of video games Hailey brother is having will be equal to 93.
What is an equation?Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.
This relationship is illustrated by the left and right expressions being equal to one another. The left-hand side equals the right-hand side is a basic, straightforward equation.
As per the information given in the question,
Stuffed animals = 5 × dolls (i)
Video games = 18 + stuffed animals (ii)
Total number of dolls Hailey has is 15
Then, put this in equation (i)
Stuffed animals = 5 × 15
= 75
Now, put 75 in equation (ii)
Video games = 18 + 75
= 93
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During a scuba dive, Lainey descended to a point 19 feet below the ocean surface. She continued her descent at a rate of 25 feet per minute. Write an inequality you could solve to find the number of minutes she can continue to descend if she does not want to reach a point more than 144 feet below the ocean surface. Use the variable t for time.
Answer:
[tex]25t+19\leq 144[/tex]
[tex]t\leq5[/tex]
The number of minutes she can continue to descend if she does not want to reach a point more than 144 feet below the ocean surface is at most 5 minutes.
Step-by-step explanation:
Given:
Initial depth of the scuba dive = 19 ft
Rate of descent = 25 ft/min
Maximum depth to be reached = 144 ft
Now, after 't' minutes, the depth reached by the scuba dive is equal to the sum of the initial depth and the depth covered in 't' minutes moving at the given rate.
Framing in equation form, we get:
Total depth = Initial Depth + Rate of descent × Time
Total depth = [tex]19+25t[/tex]
Now, as per question, the total depth should not be more than 144 feet. So,
[tex]\textrm{Total depth}\leq 144\ ft\\\\19+25t\leq 144\\\\or\ 25t+19\leq 144[/tex]
Solving the above inequality for time 't', we get:
[tex]25t+19\leq 144\\\\25t\leq 144-19\\\\25t\leq 125\\\\t\leq \frac{125}{25}\\\\t\leq 5\ min[/tex]
Therefore, the number of minutes she can continue to descend if she does not want to reach a point more than 144 feet below the ocean surface is at most 5 minutes.
The ratio of the lengths of a triangle are 4:6:9 and its perimeter is 57cm. Find the length of the shortest side.
Answer:12cm
Step-by-step explanation:
The ratio are:
4:6:9
Perimeter=57cm
Let a be the unknown value.
The triangle has three sides: 4a, 6a, 9a
Perimeter=The sum of three sides
57=4a + 6a +9a
57=19a
Divide both side by 19
a=57/19
a=3
Therefore
4a:6a:9a
4×3:6×3:9×3
12:18:27
So the shortest side is 12cm
Final answer:
The shortest side of the triangle is found by setting up the sides as a proportion with a common multiplier and using the given perimeter to solve for it. Upon finding the multiplier, we multiply it by the smallest ratio to find the shortest side, which is 12 cm.
Explanation:
To solve for the length of the shortest side of the triangle, we should initially set up the ratio of the sides as 4x:6x:9x, where x is the common multiplier for each side of the triangle. Since the perimeter of the triangle is given as 57 cm, we can write the equation 4x + 6x + 9x = 57 to represent the sum of the sides of the triangle. This simplifies to 19x = 57, and by dividing both sides of the equation by 19, we find that x = 3. Therefore, the shortest side of the triangle, represented by 4x, will be 4 × 3 = 12 cm.
What is the range of the following function?
Answer:
-4 ≤ y < ∞
Step-by-step explanation:
The graph shows the range (vertical extent) has a minimum value of -4, and includes all y-values greater than or equal to that value.
In interval notation, the range is [-4, ∞).
Your Jeep YJ is in need of new leaf spring eyebolts and bushings. All 4 springs require a total of 16, 2 piece bushings. How many spring eyebolts will you need ?
Answer:
8
Step-by-step explanation:
We assume there are 2 bushings per bolt, and 2 bolts per spring. Whether you count springs and multiply by 2, or count bushings and divide by 2, you get 8 eyebolts.
_____
Comment on the question
There are conceivable mechanical arrangements in which the number of eyebolts might be some other number. There is not actually enough information here to properly answer the question.
To replace all the bushings in a Jeep YJ, a total of 8 leaf spring eyebolts would be required considering each spring uses two eyebolts.
Explanation:To be able to replace all the bushings in your Jeep YJ's leaf springs, you need to first establish how many leaf spring eyebolts you will need. In a standard leaf spring setup, each end of the spring (both front and back) is held in place by a leaf spring eyebolt. Since you have 4 springs and each spring uses 2 leaf spring eyebolts (one for the front leaf spring eye and one for the back), you'll require a total of 8 leaf spring eyebolts for your Jeep YJ.
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There are 416 students who are enrolled in an introductory Chinese course. If there are eight boys to every five girls, how many boys are in the course?
Answer:
256 boys.
Step-by-step explanation:
Let x represent number of boys in the course.
We have been given that there are 416 students who are enrolled in an introductory Chinese course. There are eight boys to every five girls.
We will use proportions to find the total number of boys in the course.
Since there are eight boys to every five girls, so there are 8 boys for 13 (8+5) total students.
[tex]\frac{\text{Boys}}{\text{Total students}}=\frac{8}{13}[/tex]
Upon substituting our values in above equation, we will get:
[tex]\frac{x}{416}=\frac{8}{13}[/tex]
Multiply both sides by 416:
[tex]\frac{x}{416}*416=\frac{8}{13}*416[/tex]
[tex]x=8*32[/tex]
[tex]x=256[/tex]
Therefore, there are 256 boys in the course.
Final answer:
To calculate the number of boys in a class with a given ratio of boys to girls, divide the total number of students by the sum of the parts of the ratio to find one part's value, then multiply by the number of parts for boys. In this case, with 416 students and a ratio of 8 boys to 5 girls, there are 256 boys enrolled in the course.
Explanation:
Calculating the Number of Boys and Girls in a Class
To find the number of boys in an introductory Chinese course with a ratio of eight boys to every five girls, we can use the concept of ratios and proportions. Given that there are 416 students enrolled in the course, we first add the parts of the ratio together, which gives us 8 (boys) + 5 (girls) = 13 parts in total. The number of boys can then be calculated as follows:
Divide the total number of students by the total number of parts to find the value of one part: 416 students ÷ 13 parts = 32 students per part.
Multiply the value of one part by the number of parts attributed to the boys: 32 students per part × 8 parts (for boys) = 256 boys.
Therefore, based on the ratio provided, there are 256 boys enrolled in the introductory Chinese course.
Billy and Ken, the school's cross-country stars, were each running at cross-country practice. Billy was going to run 3/4 of the training course, and Ken was going to run 1/2 of the course. However, during practice it started raining, so they could not finish their run. Billy had finished 1/3 of his run, while Ken had finished 1/2 of his run. Which cross-country star ran the furthest?
Final answer:
Billy and Ken both ran 1/4 of the training course before it started raining, making the distance they ran identical; neither ran further than the other.
Explanation:
Comparison of Distances Run by Billy and Ken
To determine which cross-country star, Billy or Ken, ran the furthest, we need to calculate the fractions of the total distance each one ran. Firstly, we'll look at Billy. Billy intended to run 3/4 of the course and finished 1/3 of his planned run. Therefore, the actual distance Billy ran is (3/4) × (1/3).
Now, for Ken. Ken planned to run 1/2 of the course and completed 1/2 of his intended distance. Hence, Ken's actual distance is (1/2) × (1/2). Calculating both distances:
Billy's distance: (3/4) × (1/3) = 1/4
Ken's distance: (1/2) × (1/2) = 1/4
Both Billy and Ken ran 1/4 of the total training course, hence they ran the same distance before it started raining. Therefore, neither ran further than the other.
HELPHELP
A ferris wheel is 30 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t).
Answer:
h = f(t) = -15cos((π/5)t) +20
Step-by-step explanation:
If you like, you can make a little table of positions:
(t, h) = (0, 5), (5, 35)
Since the wheel is at an extreme position at t=0, a cosine function is an appropriate model:
h = Acos(kt) +C
The amplitude A of the function is half the difference between the t=0 value and the t=5 value:
A = (1/2)(5 -35) = -15
The midline value C is the average of the maximum and minimum:
C = (1/2)(5 + 35) = 20
The factor k satisfies the relation ...
k = 2π/period = 2π/10 = π/5
So, the function can be written as ...
h = f(t) = -15cos((π/5)t) +20
The equation of the function of the height in meters above the ground t minutes after the wheel begins to turn, is presented as follows;
[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]
The process for finding the above function is as follows;
The given parameters of the Ferris wheel are;
The diameter of the Ferris wheel, D = 30 meters
The height of the Ferris wheel above the ground, d = 5 meters
The level of loading the platform = The six o'clock position
The time it takes the wheel to make one full revolution, T = 10 minutes
The required parameter;
The function which gives the height, h, of the Ferris wheel = f(t)
Method;
The equation that can be used to model the height of the Ferris wheel is the sine function which is presented as follows;
y = A·sin[k·(t - b)] + c
Where;
A = The amplitude = (highest point - Lowest point)/2
∴ A = (35 - 5)/2 = 15
The period, T = 2·π/k
∴ 10 minutes = 2·π/k
k = 2·π/10 = π/5
k = π/5
At the starting point, the Ferris wheel is at the lowest point, and t = 0, we have;
The sine function is at the lowest point when k·(t - b) = -π/2
Therefore, we get;
π/5·(0 - b) = -π/2
-b = -π/2 × (5/π) = -5/2 = -2.5
b = 2.5
The vertical shift, c = (Max - Min)/2 + Min = (Max + Min)/2
∴ c = (35 + 5)/2 = 20
c = 20
The equation of the Ferris wheel of the form, y = A·sin[k·(t - b)] + c, is therefore;
[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]
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When a plane flies into the wind, it can travel 3000 ml in 6 h . When it flies with the wind, it can travel the same distance in 5 h. Find the rate of the plane in still air and the rate of the wind
Answer:
plane: 550 mphwind: 50 mphStep-by-step explanation:
If p and w represent the speeds of the plane and wind, respectively, the speed into the wind is ...
p - w = (3000 mi)/(6 h) = 500 mi/h
And, the speed with the wind is ...
p + w = (3000 mi)/(5 h) = 600 mi/h
Adding these two equations gives us ...
2p = 1100 mi/h
p = 550 mi/h . . . . . . . divide by 2
Then the wind speed is ...
w = 600 mi/h - p = (600 -550) mi/h
w = 50 mi/h
The rate of the plane in still air is 550 mi/h; the rate of the wind is 50 mi/h.
Answer:
Step-by-step explanation:
let speed of plane in still air =x
speed of wind=y
(x-y)6=3000
x-y=3000/6=500 ...(1)
(x+y)5=3000
x+y=3000/5=600 ...(2)
adding (1) and (2)
2x=1100
x=1100/2=550
550 +y=600
y=600-550=50
speed of plane in still air=550 m/hr
speed of wind=50 m/hr
Functions f(x) and g(x) are defined below.
Determine where f(x) = g(x) by graphing.
A.
x = -1
B.
x = 4
C.
x = -4
D.
x = -2
Answer:
C. x = -4
Step-by-step explanation:
Graphing calculators make solving a problem like this very easy. The two functions both have the value -1 at x = -4.
The solution to f(x) = g(x) is x = -4.
Suppose a room is 5.2 m long by 4.3m wide and 2.9 m high and has an air conditioner that exchanges air at a rate of 1200 L/min. How long would it take the air conditioner to exchange the air in the room
Answer:
54 minutes
Step-by-step explanation:
From the question, we are given;
A room with dimensions 5.2 m by 4.3 m by 2.9 mThe exchange air rate is 1200 L/minWe are required to determine the time taken to exchange the air in the room;
First we are going to determine the volume of the room;
Volume of the room = length × width × height
= 5.2 m × 4.3 m × 2.9 m
= 64.844 m³
Then we should know, that 1 m³ = 1000 L
Therefore, we can convert the volume of the room into L
= 64.844 m³ × 1000 L
= 64,844 L
But, the rate is 1200 L/min
Thus, time = Volume ÷ rate
= 64,844 L ÷ 1200 L/min
= 54.0367 minutes
= 54 minutes
Therefore, it would take approximately 54 minutes
Jackson has 1 3/8 kg of fertilizar. He used some to fertilizar a flower bed, and he only had 2/3 kg left. How much fertilizar was used in the flower bed
Answer:
17/24
Step-by-step explanation:
11/8 becomes 33/24
2/3 becomes 16/24
(33-16)/24 = 17/24