Answer:
[tex]R = \dfrac{16}{5}t + \dfrac{1}{5}[/tex]
where R is the R-value and t is the thickness in inches.
Step-by-step explanation:
We are given the following in the question:
The R-value of fiberglass insulation and f its thickness have a linear relation.
Let R be the R-value and t be the thickness. Then, the equation can be written as:
[tex]R = at + b[/tex]
where a and b are constants.
When t = 3.5, R = 11
[tex]11 = 3.5a + b[/tex]
When t = 6, R = 19
[tex]19 = 6a + b[/tex]
Solving the two equation, using the elimination method, we have,
[tex]19-11 = 6a + b-(3.5a + b)\\8 = 2.5a\\\\\Rightarrow a = \dfrac{16}{5}\\\\11 = 6(\dfrac{16}{5}) + b\\\\\Rightarrow b = \dfrac{1}{5}[/tex]
Thus, the linear relationship is given by:
[tex]R = \dfrac{16}{5}t + \dfrac{1}{5}[/tex]
where R is the R-value and t is the thickness in inches.
To find a linear equation (R = mt + b) that represents the R-value of fiberglass insulation as a function of its thickness, we calculate a slope (m) of 3.2 using two known measurements. We then solve for a y-intercept (b), which is -0.5. Therefore, the equation is R = 3.2t - 0.5.
Explanation:In order to derive the equation that gives the R-value as a function of the thickness, we need to use the concept of linear equations, specifically slope-intercept form (y = mx + b). First, we need to find the slope (m). Since we know two points on the line (3 ½ , 11) and (6 , 19), we calculate the slope as: m = (19-11) / (6 - 3½) = 8 / 2½ = 3.2. The y-intercept (b) is the R-value when the thickness (t) is 0. To find it, we use one of our points and the slope in the equation y = mx + b, substituting to solve for b: 11 = 3.2 * 3½ + b, hence, b = -0.5. The equation is therefore: R = 3.2t - 0.5.
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The ball's height (in meters above the ground), xxx seconds after Antoine threw it, is modeled by: h(x)=-2x^2+4x+16 How many seconds after being thrown will the ball hit the ground? seconds
Answer:
Step-by-step explanation:
If we are given the quadratic model, we know that the height of anything on the ground is 0 meters, so that model becomes
[tex]0=-2t^2+4t+16[/tex] I changed your x's to t's since t is for time.
Now we need to factor that quadratic and solve for time:
[tex]0=-2(t^2-2t-8)[/tex]
Since -2 definitely does not equal 0, then
[tex]t^2-2t-8=0[/tex]
Factor that and find t values of 4 and -2. Again, we know that time will never be negative, so
t = 4 seconds.
Find the vehicle's acceleration when its final velocity is measured as 45 m/s, the initial initial velocity was 0.0 m/s and the time elapsed was 8.5 seconds.
Answer:
Acceleration (a) [tex]=5.29\ m\./s^2[/tex]
Step-by-step explanation:
Given that
final velocity (v) [tex]=45\ m/s[/tex]
Initial velocity (u) [tex]=0\ m/s[/tex]
Time (t) [tex]=8.5\ seconds[/tex]
Then
[tex]v=u+at\ ...................................... (1)[/tex]
Where [tex]a[/tex] is acceleration.
put the value in equation [tex](1)[/tex]
[tex]45=0+a\times8.5\\a\times8.5=45[/tex]
dividing [tex]8.5[/tex] both sides
[tex]\frac{a\times8.5}{8.5}=\frac{45}{8.5}\\\\a=\frac{45}{8.5}\\\\a=\frac{450}{85}\\\\a=5.29\ m/s^2[/tex]
A local improvement store sells different sized of storage sheds. The most expensive shed has a footprint that is 15 feet wide by 21 feet long, The least expensive has a footprint that is 10 feet wide by 14 feet long. Are the footprints of the two sheds similar? If so, tell whether the footprint of the least expensive shed in a enlargement or a reduction, and find the scale factor from the most expensive shed to the least expensive shed/
Answer:
Question 1: Are the footprints of the two sheds similar?
Yes, the two sheds are similar.Question 2: Tell whether the footprint of the least expensive shed is an enlargement or a reduction,
It is a reductionQuestion 3: Find the scale factor from the most expensive shed to the least expensive shed
The scale factor is 3/2Explanation:
Question 1: Are the footprints of the two sheds similar?
The footprints of the two sheds will be similar if their measures are proportional.
The ratio of the measures of the footprint of the most expensive shed is:
width/length = 15 feet / 21 feet = 5 / 7The ratio of the measures of the footprint of the least expensive shed is:
width/length = 10 feet / 14 feet = 5 / 7Since, the two ratios are equal, you conclude that the corresponding dimensions are proportional and the two sheds are similar.
Question 2: Tell whether the footprint of the least expensive shed is an enlargement or a reduction.
A reduction is a similar transformation (the image and the preimage are similar) that maps the original figure into a smaller one.
Since the dimensions of the foot print of the least expensive shed, 10 feet wide by 14 feet long, are smaller than the dimensions of the most expensive shed, 15 feet wide by 21 feet long the you conclude that the former is a reduction of the latter.
Question 3: Find the scale factor from the most expensive shed to the least expensive shed.
To find the scale factor from the most expensive shed to the least expensive shed, you divide the measures of the corresponding dimensions. You can do it either with the widths or with the lengths.
Using the widths, you get:
width of the foot print of the most expensive shed / width of the foot print of the least expensive transformation15 feet / 10 feet = 3/2.That means that the scale factor from the most expensive shed to the least expensive shed is 3/2.
Using the lenghts, you should obtain the same scale factor:
length of the foot print of the most expensive shed / length of the foot print of the least expensive transformation21 feet / 14 feet = 3/2. Such as expected.Answer:i dont knowStep-by-step explanation:
PLEASE HELP ASAP!!! I NEED CORRECT ANSWERS ONLY PLEASE!!!
Find m∠H.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠H = °
Answer:
Step-by-step explanation:
Triangle GHI is a right angle triangle.
From the given right angle triangle,
GH represents the hypotenuse of the right angle triangle.
With m∠H as the reference angle,
HI represents the adjacent side of the right angle triangle.
GI represents the opposite side of the right angle triangle.
To determine m∠H, we would apply
the Sine trigonometric ratio.
Sin θ = adjacent side/hypotenuse. Therefore,
Sin m∠H = 7/10 = 0.7
m∠H = Sin^-1(0.7)
m∠H = 44.4° to the nearest tenth.
Answer: gayness
Step-by-step explanation:
Carmen has money in her bank account. Each week she withdraws the same amount of money from her account. Write an equation that relates b, her account balance after w weeks. 1week=$825 2weeks=$750 3weeks=$675 4weeks=$600
Step-by-step explanation:
0
A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 5 % of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 15 % of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 12 % of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.
a. What is the probability that the driver is incorrectly classified as being over the limit?
b. What is the probability that the driver is correctly classified as being over the limit?
c. Find the probability that the driver gives a breathalyser test reading that is over the limit.
d. Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.
It can be deduced that the probability that the driver is incorrectly classified is 0.044.
How to calculate probabilityFrom the information, 12 % of drivers are above the legal alcohol limit. Those below the legal alcohol limit will be:
= 1 - 0.12 = 0.88
Therefore, the probability that the driver is incorrectly classified as being over the limit will be:
= 0.05 × 0.88
= 0.044.
The the probability that the driver is correctly classified as being over the limit will be:
= P(B/A) × P(A)
= 0.85 × 0.12
= 0.102
The probability that the driver gives a breathalyser test reading that is over the limit will be:
= 0.85 × 0.12 × 0.05 × 0.88
= 0.146
Lastly, the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit will be:
= (0.88 × 0.95) / (0.88 × 0.95 + 0.12 × 0.15)
= 0.98
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The probability of incorrect classification is 0.0440, correct classification is 0.1020, a positive reading is 0.1460, and the driver being under the limit given a negative reading is 0.9299.
The breathalyser test has probabilities associated with false positives and false negatives. Let's analyse the specific questions using these probabilities.
a. Probability that the driver is incorrectly classified as being over the limit:
This occurs when a driver is under the limit but gets a false positive result. The probability of a driver being under the limit is 1 - 0.12 = 0.88. The probability of a false positive is 0.05. Hence, the probability is 0.88 * 0.05 = 0.0440.
b. Probability that the driver is correctly classified as being over the limit:
The probability of a driver being over the limit is 0.12. The probability of correctly identifying such a driver is 1 - 0.15 = 0.85. Hence, the probability is 0.12 * 0.85 = 0.1020.
c. Probability that the driver gives a breathalyser test reading that is over the limit:
This includes both true positives (driver over the limit and test positive) and false positives (driver under the limit and test positive). Hence, the probability is 0.12 * 0.85 + 0.88 * 0.05 = 0.102 + 0.044 = 0.1460.
d. Probability that the driver is under the legal limit, given the breathalyser reading is also below the limit:
This uses Bayes' theorem. P(Under limit | Test negative) = P(Test negative | Under limit) * P(Under limit) / P(Test negative).
Here, P(Test negative) = 0.88 * 0.95 + 0.12 * 0.15 = 0.836. Hence, P(Under limit | Test negative) = 0.836 / 0.880 = 0.95. So the probability is 0.836 * 0.95 / 0.854 = 0.9299 approximately.
Thus, the probability of incorrect classification is 0.0440, correct classification is 0.1020, a positive reading is 0.1460, and the driver being under the limit given a negative reading is 0.9299.
Point P is located at (2, 2) and point T is located at (7, 17).
What are the coordinates of the point that partitions the directed line segment PT in a 3:2 ratio?
Use the section formula and show values for: m: n, Point 1, Point 2, and ALL work to find coordinates of
partitioning point.
Answer:
The coordinates of the point that partitions the directed line segment PT in a 3:2 ratio will be (5, 11).
Step-by-step explanation:
Given the points
P(2, 2) and T(7, 17)What are the coordinates of the point that partitions the directed line segment PT in a 3:2 ratio?
Let D be the Divided point of the directed line segment PT.
SECTION FORMULA
The point (x, y) which partitions the line segment of the points (x₁, y₁) and
(x₂, y₂) in a ratio [tex]m:n[/tex] will be:
[tex]\left(\frac{m\:x_2+n\:x_1}{m+n},\:\frac{m\:y_2+n\:y_1}{m+n}\right)[/tex]
Here,
x₁ = 2 y₁ = 2 x₂ = 7y₂ = 17[tex]m=3,\:n=2[/tex]Substituting the values in the above formula
[tex]D\left(x\right)=\left(\frac{3\times \:7\:+\:2\times 2}{3+2},\:\frac{3\times 17\:+\:2\times 2}{3+2}\right)[/tex]
[tex]D\left(x\right)=\left(\frac{21\:+\:4}{5},\:\frac{51\:+\:4}{5}\right)[/tex]
[tex]D\left(x\right)=\left(\frac{25}{5},\:\frac{55}{5}\right)[/tex]
As
[tex]\frac{25}{5}=5,\:\frac{55}{5}=11[/tex]
So
[tex]$D(x)=(5, 11)[/tex]
Therefore, the coordinates of the point that partitions the directed line segment PT in a 3:2 ratio will be (5, 11).
A parking lot contains 100 cars, k of which happen to be lemons. We select m of these cars at random and take them for a test drive. Find the probability that n of the cars tested turn out to be lemons.
Answer:
The probability that n of the cars tested turn out to be lemons is [tex]{m\choose n}\times\frac{k^{m}(100-k)^{(m-n)}}{100^{m}}[/tex].
Step-by-step explanation:
Let X = a car is lemon
There are 100 cars in a parking lot.
The probability that a car is a lemon is:
[tex]P (A\ car\ is\ lemons) = \frac{Number\ of\ cars\ that\ are\ lemons}{Total\ number\ of\ cars} =\frac{k}{100}[/tex]
The random variable [tex]X\sim Bin (100, \frac{k}{100})[/tex]
The probability function of a Binomial distribution is:
[tex]P(X=a)={a\choose b}\times p^{b}\times (1-p)^{a-b}[/tex]
Number of cars selected is a = m.
Compute the probability that out of m cars n turn out to be lemons as follows:
[tex]P(X=n)={m\choose n}\times [\frac{k}{100}] ^{n}\times (1-\frac{k}{100} )^{m-n}={m\choose n}\times\frac{k^{m}(100-k)^{(m-n)}}{100^{m}}[/tex]
Thus, the probability that out of m cars n turns out to be lemons is [tex]{m\choose n}\times\frac{k^{m}(100-k)^{(m-n)}}{100^{m}}[/tex].
What is the range of g(x) = 3|x − 1| − 1? A. (-∞, 1] B. [-1, ∞) C. [1, ∞) D. (-∞, ∞)
Answer:
B. [-1, ∞).
Step-by-step explanation:
g(x) = 3|x − 1| − 1
When x = 1 g(x) = 3(0) - 1 = -1.
As all negative vales of x will give positive values of |x - 1| then g(x) = -1 is its minimum value. The graph will be shaped like a letter V with the vertex at (1,-1).
Therefore the range is [-1, ∞).
The range of g(x)=3|x-1|-1 is [-1,∞) i.e. option B is correct.
What is range?The set of all the outputs of a function is known as the range of the function .
According to the given question
we have,
A function, g(x) = 3|x-1| - 1
Lets, find the value of g(x) for the different values of "x"
when,
x=0 ⇒ g(0) = -1
x=1 ⇒ g(1) = -1
x=2 ⇒g(2) = 2
Similarly, we can check for the negative values of x.
So, for all the negative values of x we will gives only positive values for g(x) and only at x=0, g(x) = -1 , which is its minimum value .
⇒ The range of given function g(x) is {-1,∞).
Hence , option B is correct.
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To earn a “B” in Geometry, Abby must get a grade average of at least 80%. Abby’s test scores are 85, 90, and 75. Her final exam score counts as three test scores in the average. Abby got a grade of 78 on the final exam. What was her final grade average? Did she get a “B”?
Abby's final grade average is 69.14, which is less than 80%. Therefore, she did not get a "B".
Explanation:To calculate Abby's final grade average, we first need to find the average of her three test scores: 85 + 90 + 75 = 250. Then we need to add her final exam score, which counts as three test scores. 78 x 3 = 234. So the total score for Abby's tests and final exam is 250 + 234 = 484. Since there are a total of 7 test scores (3 tests + 3 tests for the final exam + 1 final exam), we divide the total score by 7 to get 484/7 = 69.14. Abby's final grade average is 69.14, which is less than 80%, so she did not get a "B".
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Jerome is painting a rectangular toolbox that is 20 inches by 10 inches by 8 inches. A tube of paint covers 300 square inches. What is the surface area of the toolbox?
Answer:
880 in³
Step-by-step explanation:
L = 20, W = 10, H = 8
2(h × W) + 2(h × L) + 2(W × L)
= 2(8*10) + 2(8*20) + 2(10*20)
= 2(80) + 2(160) + 2(200)
= 160 + 320 + 400
= 880 in³
You purchase a new car for $17,000 and are able to acquire a loan because of your excellent credit score. How much is the total interest and insurance per month if you use the Whome Insurance Company from question 3 for your insurance coverage and don't qualify for the safe driver discount? Credit APR (%) Excellent 5.90 Good 6.35 Average 6.85 Fair 7.40 Poor 8.15 (1 point) $95.39 $150.52 $142.64 $162.33
Answer:
D) 162.33
Step-by-step explanation:
The monthly interest on a $17,000 car loan with an APR of 5.90% is $83.3. The insurance cost isn't specified but intended to be actuarially fair. The total monthly interest and insurance cost would add these two amounts.
Explanation:In your situation, you are purchasing a car for $17,000 and are eligible for an APR (Annual Percentage Rate) of 5.90% due to your excellent credit score. You will also be acquiring insurance through the Whome Insurance Company. Let's calculate your monthly interest:
The monthly interest can be calculated by dividing your APR by 12. So, 5.90% divided by 12 equals 0.49%. Then, you multiply this by your outstanding balance of $17,000: $17,000 * 0.49% = $83.3 interest per month.
In the context of car insurance, it's mentioned that an 'actuarially fair' model would have those with a higher risk paying more. As you haven't qualified for the safe driver discount, you could be classified in a higher risk group, thus paying a higher premium. The exact figure isn't provided in the question, but it would add to your monthly interest to determine total monthly expenses.
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Three bags of sweets weigh 6 3/4 kg. Two of them have the same weight and the third bag is heavier than each of the bags of equal weight by 1 1/5 kg. Find the weight of each bag.
Answer: the weight of each bags are 1.85kg, 1.85kg and 3.05 kg
Step-by-step explanation:
Total weight of the three bags is
6 3/4 kg. Converting to decimal, it becomes 6.75 kg.
Two of them have the same weight. Let x represent the bags of equal weight. Their total weight would be 2x. The third bag is heavier than each of the bags of equal weight by 1 1/5 kg. Converting to decimal, it becomes 1.2kg. It means that the weight of the third bag would be
x + 1.2
Therefore, for the three bags,
2x + x + 1.2 = 6.75
3x = 6.75 - 1.2 = 5.55
x = 5.55/3 = 1.85
Weight of the third bag is
1.85 + 1.2 = 3.05kg
To solve for the weights of the bags, we set up an equation and find that the weight of each of the lighter bags is 1 17/20 kg, and the weight of the heavier bag is 3 1/20 kg.
Explanation:The question asks us to calculate the weight of each bag of sweets when given the total weight and the additional weight of the heavier bag. Let's denote the weight of the two equal bags as 'x' kilograms each. According to the problem, the third bag weighs 'x + 1 1/5' kg, which is heavier by 1 1/5 kg than each of the other two bags. The total weight of the three bags is 6 3/4 kg.
Starting with this information, we can set up an equation to find the value of 'x': 2x + (x + 1 1/5 kg) = 6 3/4 kg. To solve for 'x', first simplify the right side of the equation by converting the mixed fraction to an improper fraction: 6 3/4 kg = 27/4 kg. Now, convert 1 1/5 kg to an improper fraction as well: 1 1/5 kg = 6/5 kg. The equation now is: 2x + x + 6/5 = 27/4 kg.
Combining like terms, we have: 3x = 27/4 kg - 6/5 kg. To combine these fractions, we need a common denominator, which would be 20 in this case. This changes the equation to: 3x = (27/4) * (5/5) - (6/5) * (4/4) = 135/20 kg - 24/20 kg = 111/20 kg. To find 'x', divide both sides by 3: x = (111/20 kg) / 3 = (111/20) * (1/3) kg = 37/20 kg = 1 17/20 kg. This is the weight of each of the lighter bags.
Subsequently, the weight of the heavier bag can be found by adding the additional weight: x + 6/5 kg = (37/20 kg) + (6/5 kg) = (37/20) + (24/20) = 61/20 kg = 3 1/20 kg.
Therefore, the weight of the two bags with equal weight is 1 17/20 kg each, and the weight of the heavier bag is 3 1/20 kg.
In reference to the diagram below, Juan claims that the area of circle B is twice the area of circle A since it's radius is twice as big.
Circle A radius = 2
Circle B radius = 4
Explain why Juan is mistaken in his conclusion about the area of circle B. In writing your response, be sure to explain what happens to the radius when it is squared. Explain the steps in finding the area of a circle.
Answer:
circle b
Step-by-step explanation:
Answer:
Step-by-step explanation:
amosc: marcos62280 ( ;
What is the slope of the line passing through the points (−1, −7) and (−9, −2) ?
[tex]\bf (\stackrel{x_1}{-1}~,~\stackrel{y_1}{-7})\qquad (\stackrel{x_2}{-9}~,~\stackrel{y_2}{-2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-2}-\stackrel{y1}{(-7)}}}{\underset{run} {\underset{x_2}{-9}-\underset{x_1}{(-1)}}}\implies \cfrac{-2+7}{-9+1}\implies -\cfrac{5}{8}[/tex]
Answer:
The answer to your question is m = [tex]\frac{5}{8}[/tex]
Step-by-step explanation:
Data
A (-1, -7)
B (-9, -2)
Formula
slope = m = [tex]\frac{y2 - y1}{x2 - x1}[/tex]
Use the slope formula to find the answer. Just substitute the values and simplify them.
Substitution
x1 = -1
x2 = -9
y1 = -7
y2 = -2
m = [tex]\frac{-2 + 7}{-9 + 1}[/tex]
Simplification
m =- [tex]\frac{5}{8}[/tex]
Result
m =- [tex]\frac{5}{8}[/tex]
A cable that weighs 2 lbyft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.
Answer:
W = 650000 ftlb
Step-by-step explanation:
Parameters given:
Weight of the cable = 2 lb/ft
Weight of the coal = 800 lb
Depth of the shaft, x = 500 ft
The total work done in this system is given by the sum of the work done on the coal and the work done on the shaft.
WORK DONE ON THE COAL
The work done on the coal is given as:
[tex]W_{coal}[/tex] = Weight of coal * distance
[tex]W_{coal}[/tex] = 800 * 500
[tex]W_{coal}[/tex] = 400000 ftlb
WORK DONE ON THE SHAFT
If we partition the cable into several segments of depth Δx, we can find the work done on the cable.
The work done on the shaft is given as:
[tex]W_{shaft}[/tex] = 2Δx * x
[tex]W_{shaft}[/tex] = 2xΔx
=> [tex]W_{shaft}[/tex] = [tex]\int\limits^{500}_{0} {2x} \, dx[/tex]
[tex]W_{shaft}[/tex] = [tex]\frac{2x^{2}}{2} \left \{ {{500} \atop {0}} \right.[/tex]
[tex]W_{shaft}[/tex] = [tex]500^{2} - 0^{2}[/tex]
[tex]W_{shaft}[/tex] = 250000 ftlb
Therefore, total work done is:
[tex]W_{total} = 250000 + 400000[/tex]
[tex]W_{total} = 650000 ftlb[/tex]
The work done to lift the coal up the mine shaft is 900,000 ft-lbf.
To find the work done, we can use the formula:
Work = Force x Distance
First, we need to calculate the force. The cable weighs 2 lbf/ft, so the total weight of the cable is 2 lbf/ft x 500 ft = 1000 lbf. The force to lift the coal is 800 lbf. Therefore, the total force is 1000 lbf + 800 lbf = 1800 lbf.
Next, we can calculate the work done by multiplying the force by the distance: Work = 1800 lbf x 500 ft = 900,000 ft-lbf.
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Jamie wants to treat some friends to lunch.He has $40 and knows that lunch will cost about $7 per person,p. How many people can Jamie buy lunch for? Write and solve an inequality to represent this situation
Answer:
[tex]7p\leq 40[/tex]
Jamie can buy lunch for 5 person.
Step-by-step explanation:
Given:
Jamie has $40.
Lunch cost per person = $7
Solution:
Jamie wants to treat some friends to lunch.He has $40 and knows that lunch will cost about $7 per person, p.
So, an inequality to represent this situation is written as:
[tex](Lunch\ cost\ per\ person)p\leq Total\ amount[/tex]
[tex]7p\leq 40[/tex]
[tex]p\leq \frac{40}{7}[/tex]
[tex]p\leq 5.71[/tex]
Hence, the answer is 5
Therefore, Jamie can buy lunch for 5 persons.
Suppose the position of an object moving in a straight line is given by s(t) = t^2+5t+2.
a) Find the average velocity between t = 1 and t=5
b) Find the instantaneous Velocity t=1 and t=5
Answer:
Step-by-step explanation:
Finding the average velocity for time t=1 and t= 5
For t^2+ 5t+ 2
Putting 5 and 1 in to the equation
= (5^2)-(1^2) + 5(5-1)+ 2
= 25-1+ 5*4+2
=24+20+2
=46m/s
For instantaneous velocity
ds/dt is found
Which gives 2t+ 5
Putting t= 5 15
Putting t= 1. 7
Subtracting the answer 8m/s.
the average velocity between [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex] is 11 units/time, and the instantaneous velocities at [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex] are 7 units/time and 15 units/time, respectively.
To find the average velocity between t = 1 and t = 5, we use the formula for average velocity:
[tex]\[ \text{Average Velocity} = \frac{\text{Change in Position}}{\text{Change in Time}} \][/tex]
a) Average velocity between t = 1 and t = 5 :
First, we find the position at t = 1 and t = 5 by plugging the values into the position function [tex]\( s(t) = t^2 + 5t + 2 \)[/tex]:
At t = 1:
[tex]\[ s(1) = (1)^2 + 5(1) + 2 = 1 + 5 + 2 = 8 \][/tex]
At t = 5:
[tex]\[ s(5) = (5)^2 + 5(5) + 2 = 25 + 25 + 2 = 52 \][/tex]
Now, we calculate the change in position:
[tex]\[ \text{Change in Position} = s(5) - s(1) = 52 - 8 = 44 \][/tex]
The change in time is \( 5 - 1 = 4 \).
[tex]\[ \text{Average Velocity} = \frac{44}{4} = 11 \, \text{units/time} \][/tex]
b) Instantaneous velocity at t = 1 and t = 5 :
The instantaneous velocity at any time t is given by the derivative of the position function [tex]\( s(t) \)[/tex] with respect to t, denoted as [tex]\( s'(t) \)[/tex] or [tex]\( \frac{ds}{dt} \)[/tex].
[tex]\[ s(t) = t^2 + 5t + 2 \][/tex]
Taking the derivative with respect to [tex]\( t \)[/tex]:
[tex]\[ s'(t) = \frac{ds}{dt} = 2t + 5 \][/tex]
Now, we can find the instantaneous velocities at [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex] by plugging these values into [tex]\( s'(t) \)[/tex]:
At t = 1:
[tex]\[ s'(1) = 2(1) + 5 = 2 + 5 = 7 \, \text{units/time} \][/tex]
At t = 5:
[tex]\[ s'(5) = 2(5) + 5 = 10 + 5 = 15 \, \text{units/time} \][/tex]
So, the average velocity between [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex] is 11 units/time, and the instantaneous velocities at [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex] are 7 units/time and 15 units/time, respectively.
N To be able to afford the house of their dreams, Dave and Leslie must clear $128,000 from the sale of their first house. If they must pay $780 in closing costs and 6% of the selling price for the sales commission, then what is the minimum selling price for which they will get $128,000?
Answer:
the selling price of the house is estimated as $137,000
Step-by-step explanation:
Given data:
clear amount is $128,000
closing cost is $780
sales commission rate is 6%
let S be the selling price of house
from the information given in the question we have following relation
[tex]128,000 \leq S - 0.06S - 780[/tex]
[tex]128,000 \leq 0.94S - 780[/tex]
adding 780 on both side, we get
[tex]128,000 - 780 \leq 0.94S[/tex]
divide the both side by 0.94, we get
[tex]137,000 \leq S[/tex]
therefor the selling price of the house is estimated as $137,000
Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected . Find the probability of selecting a black card and selecting a red card. The probability of selecting a black card and then selecting a red card is
(A) Probability of the black card is 1 /2.
(B) Probability of a red card is 26/51.
(C) Probability of selecting a black card and then selecting a red card is 13/51.
Probability can be defined as the ratio of favorable outcomes to the total number of events.
Here,
Total number of cards in the deck = 52
Total number of black cards = 26
Total number of red cards = 26
Probability of selecting a black card = 26 / 52 = 1 / 2
Now, the total number of card become after selecting 1 black card = 51
Probability of selecting a red card = 26 / 51
Now,
The probability of selecting a black card and then selecting a red card is,
= 1 / 2 * 26 / 51
= 13 / 51
Thus, (A) the Probability of a black card is 1 /2.
(B) Probability of a red card is 26/51.
(C) Probability of selecting a black card and then selecting a red card is 13/51.
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The probability of selecting a black card first and then a red card is [tex]\(\frac{13}{51}\).[/tex]
To find the probability of selecting a black card and then selecting a red card from a standard deck of 52 playing cards without replacement, we need to follow these steps:
1. Calculate the probability of selecting a black card first:
- There are 26 black cards in a standard deck (13 spades and 13 clubs).
- The probability of selecting a black card first is:
[tex]\[ P(\text{Black card first}) = \frac{26}{52} = \frac{1}{2} \][/tex]
2. Calculate the probability of selecting a red card second, given that the first card was black:
- After drawing the first black card, there are 51 cards left in the deck.
- There are still 26 red cards in the deck (since the black card was not replaced).
- The probability of selecting a red card given that the first card was black is:
[tex]\[ P(\text{Red card second} \mid \text{Black card first}) = \frac{26}{51} \][/tex]
3. Calculate the combined probability of both events occurring:
- The probability of both events happening (selecting a black card first and then a red card) is the product of the individual probabilities:
[tex]\[ P(\text{Black card first and Red card second}) = P(\text{Black card first}) \times P(\text{Red card second} \mid \text{Black card first}) \][/tex]
Substituting the values we found:
[tex]\[ P(\text{Black card first and Red card second}) = \left(\frac{1}{2}\right) \times \left(\frac{26}{51}\right) = \frac{26}{102} = \frac{13}{51} \][/tex]
Therefore, the probability of selecting a black card first and then a red card is [tex]\(\frac{13}{51}\).[/tex]
You are painting your two car garage, including the door. The walls measure 12 feet by 12 feet, with a 10-foot ceiling. How many gallons of paint will you buy if it covers 400 square feet per gallon?
Answer:
1.2 gallons
Step-by-step explanation:
We assume that each room is rectangular with four walls and no doors. If the dimensions are length l, by width w with height h, then there are two walls with area l*h and two walls with w*h.
The total area of the room is thus: 2lh + 2wh = 2h(l+w).
l = 12ft, w = 12ft, h = 10ft (all given)
Area of walls = 2h(l+w) = 2*10(12+12)
Area = 2*10*24 = 480 square feet
Each gallon of paint covers 400 square feet. So, you will need to buy
480 / 400 = 1.2 gallons
You will need to buy 2 gallons
A random sample of the actual weight of 5-lb bags of mulch produces a mean of 4.963 lb and a standard deviation of 0.067 lb. If n=50, which of the following will give a 95% confidence interval for the mean weight (in pounds) of the mulch produced by this company?
A) 4.963±0.016.
B) 4.963±0.019.
C) 4.963±0.067.
D) 4.963±0.009.
E) None of the above.
Answer: B) 4.963±0.019.
Step-by-step explanation:
Confidence interval for population mean ( when population standard deviation is not given) is given by :-
[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]
, where [tex]\overline{x}[/tex] = Sample mean
n= Sample size
s= sample standard deviation
t* = critical t-value.
As per given:
n= 50
Degree of freedom = n-1 =49
[tex]\overline{x}= 4.963\ lb[/tex]
s= 0.067 lb
For df = 49 and significance level of 0.05 , the critical two-tailed t-value ( from t-distribution table) is 2.010.
Now , substitute all values in the formula , we get
[tex]4.963\pm (2.010)\dfrac{0.067}{\sqrt{50}}\\\\ 4.963\pm (2.010)(0.0094752)\\\\ 4.963\pm0.019045152\approx4.963\pm0.019[/tex]
Hence, a 95% confidence interval for the mean weight (in pounds) of the mulch produced by this company is [tex]4.963\pm0.019[/tex].
Thus , the correct answer is B) 4.963±0.019.
Answer:
the correct answer is B) 4.963±0.019.
Step-by-step explanation:
Laura buys Candy the cost five dollars per pound she will buy less than 9 pounds of candy what are the possible amounts you spend on candy you see for the amount in dollars Laura will spend on candy right your answer is it a inequality
A candy jar contains 200 pieces of candy Each of which is either red white or blue there are 50 red pieces and 40 white pieces if B represents the number of blue pieces in the jar by the equation you can use to find the value of B South equation label your answer
Answer:
200-50-40=B
B =110 blue pieces
Step-by-step explanation:
Stephanie is saving money to buy a new computer. So far she has saved $200. Write an inequality to show how much she needs to save each month for the next year so she has at least $1200 to spend on the computer,then solve the inequality.
Answer:
Step-by-step explanation:
Let x represent the amount that she needs to save each month for the next year.
Stephanie is saving money to buy a new computer. So far she has saved $200. This means that the total amount that she would have saved in y months is
200 + xy
Since there are 12 months in a year,
an inequality to show how much she needs to save each month for the next year so she has at least $1200 to spend on the computer is
200 + 12x ≥ 1200
12x ≥ 1200 - 200
12x ≥ 1000
x ≥ 1000/12
x ≥ 83.33
Out of 50 students taking a midterm psychology exam, 26 answered the first of two bonus questions, 33 answered the second bonus question, and 2 didn't bother with either
To find out how many students answered at least one bonus question out of 50, we subtract the number who didn't answer any from the total, resulting in 48 students.
To determine how many students answered at least one of the bonus questions in a midterm psychology exam, we can use the principle of inclusion-exclusion.
Given that 26 students answered the first bonus question,
3 students answered the second bonus question,
2 students didn't answer either, we can find the total number of students who attempted at least one bonus question.
First, add the number of students who answered each question: 26 + 33 = 59.
However, this count includes the students who answered both questions twice.
Since the total number of students is 50, and 2 didn't answer any bonus questions, 48 students must have attempted at least one bonus question.
The apparent excess (59) suggests overlaps, indicating students answered both questions.
To find the exact number of students who answered at least one bonus question, we subtract the total number of students by those who didn't attempt any: 50 - 2 = 48.
Therefore, 48 students answered at least one of the bonus questions.
The Complete Question is:
out of 50 students taking a mid-term psychology exam, 23 answered the first of two bonus questions, 36 answered the second bonus question, and 2 didn't with either one. round your answer to the nearest percent.
Determine whether the statement makes sense or does not make sense, and explain your reasoning. I found a strong negative correlation for data relating the percentage of people in various countries who are literate and the percentage who are undernourished. I concluded that an increase in literacy causes a decrease in undernourishment.a. The statement does not make sense. There is no way for an increase in literacy to cause a decrease in undernourishment b. The statement does not make sense. Correlation is not necessarily causation c. The statement makes sense if a correlation exists between two variables it can be concluded that an increase or decrease in one variable causes an mcrease or decrease in the other d. The statement makes sense The correlation between ncreased literacy and decreased undernourishment is simply a coincidence
Answer:
b
Step-by-step explanation:
In a recent poll,only 8% of the people surveyed were againt a new bill making it mandatory to recycle. How many of he 75 people surveyed were against the bill?
Answer:
There were 6 people against the bill.
Step-by-step explanation:
Given:
In a recent poll,only 8% of the people surveyed were against a new bill making it mandatory to recycle.
Now, to find of the 75 people surveyed were against the bill.
Total number of people = 75.
Percent of the people surveyed were against the bill = 8%.
Now, to get the number of people who were against the bill:
8% of 75
[tex]=\frac{8}{100}\times 75[/tex]
[tex]=0.08\times 75[/tex]
[tex]=6.[/tex]
Therefore, there were 6 people against the bill.
A three‐digit number satisfies the following conditions: The digits are consecutive whole numbers in increasing order; the sum of each pair of digits is greater than 4 and less than 10; and the tens digit is an odd number. What is the number?
The only three-digit number meeting all the conditions is 234, where 2, 3, and 4 are consecutive numbers the tens digit is an odd number, and the sum of each pair of digits is more than 4 and less than 10.
Explanation:In solving this problem, we need to apply logic based on the given conditions. The number is a three‐digit number. The digits are consecutive whole numbers in increasing order and the tens digit is an odd number. Given the sum of each pair of digits should be more than 4 and less than 10, the only possible digits that can be put in a row and satisfy these conditions are 2, 3, and 4 or 4, 5, and 6. From these options, only 2, 3, and 4 have the tens place occupied by an odd number. So, the correct number is 234.
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Final answer:
The three-digit number that meets all the given conditions, including having consecutive digits in increasing order with an odd middle digit and each pair's sum between 4 and 10, is 345.
Explanation:
To find a three-digit number that meets the given conditions, we first identify the critical factors: The digits are consecutive whole numbers in increasing order, each pair of digits' sum must be greater than 4 and less than 10, and the tens digit is an odd number. Therefore, we proceed with a logical approach to solve this.
Since the tens digit must be odd, possible options for the tens digit are 1, 3, 5, 7, or 9.
Considering the sum of each pair must be greater than 4 and less than 10 limits our options. If we start with 1, the consecutive numbers don't satisfy the sum condition strictly. Similarly, higher starting digits will not fit the conditions.
Realizing the only sequence of three consecutive numbers, among which the middle one is odd and every pair's sum falls within the required range, is 3, 4, and 5. Hence, 345 is the only number that satisfies all the given conditions.
A force of 6 lb is required to hold a spring stretched 8 in. beyond its natural length. How much work W is done in stretching it from its natural length to 11 in. beyond its natural length?
Answer:
Work done is 1.02J
Step-by-step explanation:
Work (W) done in stretching a spring = 1/2Fe
F (force) = 6 lb = 6×4.4482N = 26.6892N
e (extension) = 11in - 8in = 3in = 3×0.0254m = 0.0762m
W = 1/2 × 26.6892 × 0.0762 = 1.02J