Answers
Step-by-step explanation:
or,[(√-x-1)+(√x+9)]^2=4^2
or,(√-x-1)^2+2√(-x-1)(x+9)+(√x+9)^2=16
or,-x-1+2√-x^2-10x-9 +x+9=16
or,2√-x^2-10x-9=8
or,√-x^2-10x-9=4
squaring on both sides
or,-x^2-10x-9=16
or,-x^2-10x=25
or,-x(x+10)=25
Either,
x=-25 or, x=15.
Related Questions
Using Normal Distribution, what is the area to the right of 0.72 under the
standard normal curve?
Answers
Answer: 0.2358
Step-by-step explanation:
Using Normal Distribution, under the standard normal curve
The area to the right of z is given by P(Z>z)=1-P(Z<z)
So, the area to the right of z= 0.72 under the standard normal curve would be:
P(Z>0.72)=1-P(z<0.72)
=1-0.7642 [By using p-value table]
= 0.2358
Hence, the area to the right of z= 0.72 under the standard normal curve is 0.2358 .
The net of a triangular prism is shown below. What is the surface area of the prism? A. 128 cm^2 B. 152 cm^2 C. 176 cm^2 D. 304 cm^2
Answers
Answer:
B. 152 cm²
Step-by-step explanation:
To find the surface area using a net, do this:
Take apart the figure. We see that there are three rectangles and two triangles. Find the area of each ([tex]A=l*w[/tex]) and then add the values together:
The first rectangle on the left is the same as the one on the right.
[tex]5*8=40[/tex]
Two measures are 40 cm².
The middle rectangle is:
[tex]6*8=48[/tex]
48 cm²
The formula for the area of a triangle is [tex]A=\frac{1}{2}*b*h[/tex]:
[tex]A=\frac{1}{2}*6*4\\\\A=\frac{1*6*4}{2}\\\\A=\frac{24}{2}\\\\ A=12[/tex]
The area of the two triangles is 12 cm².
Now add the values:
[tex]40+40+48+12+12=152[/tex]
The area of the figure is 152 cm².
:Done
How many dimensions does an angle have?
Answers
Answer:
the length has dimension 1, the area has the dimension 2, the volume has dimension 3, etc. And the angle has dimension 0.
Step-by-step explanation:
Suppose P( A) = 0.60, P( B) = 0.85, and A and B are independent. The probability of the complement of the event ( A and B) is: a. .4 × .15 = .060 b. 0.40 + .15 = .55 c. 1 − (.40 + .15) = .45 d. 1 − (.6 × .85) = .490
Answers
Answer: a. 0.4 × 0.15 = 0.060
Step-by-step explanation: Probability of the complement of an event is the one that is not part of the event.
For P(A):
P(A') = 1 - 0.6
P(A') = 0.4
For P(B):
P(B') = 1 - 0.85
P(B') = 0.15
To determine probability of A' and B':
P(A' and B') = P(A')*P(B')
P(A' and B') = 0.4*0.15
P(A' and B') = 0.06
Probability of the complement of the event is 0.060
[PLEASE HELP] Consider this function, f(x) = 2X - 6.
Match each transformation of f (x) with its descriptions..
Answers
Answer:
Find answer below
Step-by-step explanation:
f(x)=2x-6
Domain of 2x-6: {solution:-∞<x<∞, interval notation: -∞, ∞}
Range of 2x-6: {solution:-∞<f(x)<∞, interval notation: -∞, ∞}
Parity of 2x-6: Neither even nor odd
Axis interception points of 2x-6: x intercepts : (3, 0) y intercepts (0, -6)
inverse of 2x-6: x/2+6/2
slope of 2x-6: m=2
Plotting : y=2x-6
if 2500 amounted to 3500 in 4 years at simple interest. Find the rate at which interest was charged
Answers
Answer:
35%
Step-by-step explanation:
[tex]Principal = 2500\\\\Simple\:Interest = 3500\\\\Time = 4 \:years\\\\Rate = ?\\\\Rate = \frac{100 \times Simple \: Interest }{Principal \times Time}\\\\Rate = \frac{100 \times 3500}{2500 \times 4} \\\\Rate = \frac{350000}{10000}\\\\ Rate = 35 \%[/tex]
[tex]S.I = \frac{PRT}{100}\\\\ 100S.I = PRT\\\\\frac{100S.I}{PT} = \frac{PRT}{PT} \\\\\frac{100S.I}{PT} = R[/tex]
Answer:
35%
Step-by-step explanation:
I REALLY HOPE I HELPED
HOPE I HELPED
PLS MARK BRAINLIEST
DESPERATELY TRYING TO LEVEL UP
✌ -ZYLYNN JADE ARDENNE
JUST A RANDOM GIRL WANTING TO HELP PEOPLE!
PEACE!
Find the area of the shaded regions:
Answers
area of Arc subtending [tex]360^{\circ}[/tex] (i.e. the whole circle) is $\pi r^2$
so area of Arc subtending $\theta^{\circ}$ is, $\frac{ \pi r^2}{360^{\circ}}\times \theta^{\circ}$
$\theta =72^{\circ}$ so the area enclosed by one such arc is $\frac{\pi (10)^272}{360}$
abd there are 2 such arcs, so double the area.
[tex] \LARGE{ \underline{ \boxed{ \rm{ \purple{Solution}}}}}[/tex]
Given:-Radius of the circle = 10 inchesAngle of each sector = 72°Number of sectors = 2To FinD:-Find the area of the shaded regions....?How to solve?For solving this question, Let's know how to find the area of a sector in a circle?
[tex] \large{ \boxed{ \rm{area \: of \: sector = \frac{\theta}{360} \times \pi {r}^{2} }}}[/tex]
Here, Θ is the angle of sector and r is the radius of the circle. So, let's solve this question.
Solution:-We have,
No. of sectors = 2Angle of sector = 72°By using formula,
⇛ Area of shaded region = 2 × Area of each sector
⇛ Area of shaded region = 2 × Θ/360° × πr²
⇛ Area of shaded region = 2 × 72°/360° × 22/7 × 10²
⇛ Area of shaded region = 2/5 × 100 × 22/7
⇛ Area of shaded region = 40 × 22/7
⇛ Area of shaded region = 880/7 inch. sq.
⇛ Area of shaded region = 125.71 inch. sq.
☄ Your Required answer is 125.71 inch. sq(approx.)
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A population has a mean and a standard deviation . Find the mean and standard deviation of a sampling distribution of sample means with sample size n. nothing (Simplify your answer.) nothing (Type an integer or decimal rounded to three decimal places as needed.)
Answers
Complete Question
A population has a mean mu μ equals = 77 and a standard deviation σ = 14. Find the mean and standard deviation of a sampling distribution of sample means with sample size n equals = 26
Answer:
The mean of sampling distribution of the sample mean ( [tex]\= x[/tex]) is [tex]\mu_{\= x } = 77[/tex]
The standard deviation of sampling distribution of the sample mean ( [tex]\= x[/tex]) is
[tex]\sigma _{\= x} = 2.746[/tex]
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 77[/tex]
The standard deviation is [tex]\sigma = 14[/tex]
The sample size is [tex]n = 26[/tex]
Generally the standard deviation of sampling distribution of the sample mean ( [tex]\= x[/tex]) is mathematically represented as
[tex]\sigma _{\= x} = \frac{ \sigma }{ \sqrt{n} }[/tex]
substituting values
[tex]\sigma _{\= x} = \frac{ 14}{ \sqrt{26} }[/tex]
[tex]\sigma _{\= x} = 2.746[/tex]
Generally the mean of sampling distribution of the sample mean ( [tex]\= x[/tex]) is equivalent to the population mean i.e
[tex]\mu_{\= x } = \mu[/tex]
[tex]\mu_{\= x } = 77[/tex]
Solve for y.
-1 = 8+3y
Simplify you answer as much as possible.
Answers
Answer:
-3
Step-by-step explanation:
[tex]8+3y = -1\\3y = -9\\y = -3[/tex]
Answer:
y = -3
Step-by-step explanation:
-1=3y+8
3y+8=-1
3y=-9
y=-3
Factor this trinomial completely. -6x^2 +26x+20
Answers
Answer:
Step-by-step explanation:
-6x²+26x+20
=-2(3x²-13x-10)
=-2(3x²-15x+2x-10)
=-2[3x(x-5)+2(x-5)]
=-2(x-5)(3x+2)
can someone help me answer this??
Answers
Answer:
hkkr
need school the long said
Answer:
That would indicate 20.0 ml
id appreciate a rating thanks XP
A company has 8 mechanics and 6 electricians. If an employee is selected at random, what is the probability that they are an electrician
Answers
Answer:
[tex]Probability = \frac{3}{7}[/tex]
Step-by-step explanation:
Given
Electrician = 6
Mechanic = 8
Required
Determine the probability of selecting an electrician
First, we need the total number of employees;
[tex]Total = n(Electrician) + n(Mechanic)[/tex]
[tex]Total = 6 + 8[/tex]
[tex]Total = 14[/tex]
Next, is to determine the required probability using the following formula;
[tex]Probability = \frac{n(Electrician)}{Total}[/tex]
[tex]Probability = \frac{6}{14}[/tex]
Divide numerator and denominator by 2
[tex]Probability = \frac{3}{7}[/tex]
Hence, the probability of selecting an electrician is 3/7
In training to run a half marathon, Jenny ran 2/5 hours on Tuesday, 11/6 hours on
Thursday, and 21/15 hours on Saturday. What is the total amount of hours that Jenny
ran this week? (Simplify your answer and state it as a mixed number.)
I
Answers
Answer:
Total hours that Jenny ran = 3.63 hours.
Step-by-step explanation:
Jenny ran on Tuesday for = 2/5 hours or 0.4 hours.
Time consumed to run on Thursday = 11/6 hours or 1.83 hours.
Time consumed to run on Saturday = 21/ 15 hours or 1.4 hours.
Here, the total hours can be calculated by just adding all the running hours. So the running hours of Tuesday, Thursday, and Saturday will be added to find the total hours.
Total hours that Jenny ran = 0.4 + 1.83 + 1.4 = 3.63 hours.
Find the point(s) on the ellipse x = 3 cost, y = sin t, 0 less than or equal to t less than or equal to 2pi closest to the point(4/3,0) (Hint: Minimize the square of the distance as a function of t.) The point(s) on the ellipse closest to the given point is(are) . (Type ordered pairs. Use a comma to separate answers as needed.)
Answers
Answer and Step-by-step explanation:
The computation of points on the ellipse is shown below:-
Distance between any point on the ellipse
[tex](3 cos t, sin t) and (\frac{4}{3},0) is\\\\ d = \sqrt{(3 cos\ t - \frac{4}{3}^2) } + (sin\ t - 0)^2\\\\ d^2 = (3 cos\ t - \frac{4}{3})^2 + sin^2 t[/tex]
To minimize
[tex]d^2, set\ f' (t) = 0\\\\2(3cos\ t - \frac{x=4}{3} ).3(-sin\ t) + 2sin\ t\ cos\ t = 0\\\\ 8 sin\ t - 16 sin\ t\ cos\ t = 0\\\\ 8 sin\ t (1 - 2 cos\ t) = 0\\\\ sin\ t = 0, cos\ t = \frac{1}{2} \\\\ t= 0, \ 0, \pi,2\pi,\frac{\pi}{3} , \frac{5\pi}{3}[/tex]
Now we create a table by applying the critical points which are shown below:
t [tex]d^{2} = (3\ cos t - \frac{4}{3})^{2} + sin^{2}t[/tex]
0 [tex]\frac{25}{9}[/tex]
[tex]\pi[/tex] [tex]\frac{169}{9}[/tex]
[tex]2\pi[/tex] [tex]\frac{25}{9}[/tex]
[tex]\frac{\pi}{3}[/tex] [tex]\frac{7}{9}[/tex]
[tex]\frac{5\pi}{3}[/tex] [tex]\frac{7}{9}[/tex]
When t = [tex]\frac{\pi}{3}[/tex], x is [tex]\frac{3}{2}[/tex] and y is [tex]\frac{\sqrt{3} }{2}[/tex]. So, the required points are [tex](\frac{3}{2},\frac{\sqrt{3} }{2})[/tex]
When t = [tex]\frac{5\pi}{3}[/tex], x is [tex]\frac{3}{2}[/tex] and y is [tex]\frac{-\sqrt{3} }{2}[/tex]. So, the required points are [tex](\frac{3}{2},\frac{-\sqrt{3} }{2})[/tex]
Jill works at a cell phone store. Jill earns $175 every week plus $45 for every phone p that she sells. if Jill makes $445 at the end of the week how many phones did she sell?
Answers
━━━━━━━☆☆━━━━━━━
▹ Answer
6 phones
▹ Step-by-Step Explanation
$445 - $175 = $270
$270 ÷ $45 = 6
6 phones
Hope this helps!
CloutAnswers ❁
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line m in the xy-plane above is to be reflected through the x-axis. if the slope of line m is 2/3,whats is the slope of the image of line m under the reflection.
Answers
Answer: The new slope is -(2/3)
Step-by-step explanation:
Ok, we know that our line can be written as:
y = (2/3)*x + b
where b is the y-intercept, and here does not really matter.
Ok, remember that if we have a point (x, y) and we reflect it over the x-axis, the new point will be (x, -y).
For our linear equation, the point (x, y) can be written as:
(x, y = (2/3)*x + b) = (x, (2/3)*x + b)
Now, after the reflection, our point is:
(x, - ( (2/3)*x + b)) = (x, -(2/3)*x - b)
Then our new line is y = -(2/3)*x - b
The new slope is -(2/3)
Find the distance between the points. Give an exact answer and an approximation to three decimal places.
TI
(S.
(3.1, 0.3) and (2.7, -4.9)
Th
(Rd
Answers
Answer:
5.215 units (rounded up to three decimal places)
Step-by-step explanation:
To find the distance between points (3.1 , 0.3) and (2.7, -4.9)
We use the Pythagoras Theorem which states that for a right triangle of sides a,b and c then;
a² + b² = c² , Where c is the hypotenuse.
In our case, the distance between the two points is the hypotenuse of triangle formed by change in y-axis and change in x-axis.
The distance (hypotenuse) squared = (-4.9 - 0.3)² + (2.7 - 3.1)² = 27.04 + 0.16 = 27.2
Hypotenuse (the distance between) = [tex]\sqrt{27.2}[/tex] = 5.215 units (rounded up to three decimal places)
GIVING OUT BRAINLIEST TO THE FIRST PERSON TO ANSWER!!
One circle has a diameter of 6 inches. A second, larger circle has a diameter that is four times the diameter of the first circle. What is the ratio of the area of the smaller circle to the larger circle?
A. 2:3
B. 1:6:4
C. 1:16
D. 1:64
Please include ALL work! <3
Answers
Answer:
The answer is option CStep-by-step explanation:
To find the ratio first find the diameter of the larger circle
Diameter of first circle = 6 inches
Diameter of second circle = 4 × diameter of the first circle
Which is
Diameter of second circle
= 4 × 6 = 24 inches
Area of a circle = πr²
r is the radius
Area of smaller circle
Diameter = 6 inches
Radius = 6 / 2 = 3 inches
Area = (3)² π = 9π in²
Area of larger circle
Diameter = 24 inches
Radius = 24 / 2 = 12 inches
Area = (12)²π = 144π in²
The ratio of the smaller circle to the larger circle is
[tex] \frac{9\pi}{144\pi} [/tex]
Reduce the fraction by 9π
That's
[tex] \frac{1}{16} [/tex]
We have the final answer as
1 : 16Hope this helps you
Answer:
C. 1:16
Step-by-step explanation:
Area of a circle is:
[tex]\pi \times {r}^{2} [/tex]
Small circle Area:
radius = diameter/2
radius = 6/2 = 3
[tex]area \: of \: a \: circle \: = \pi {3}^{2} [/tex]
a = 28.27
Large circle 4 times larger diameter
6*4 = 24
diameter = 24
r = 24/2
r = 12
[tex]a \: = \pi {12}^{2} [/tex]
a = 452.39
area of large circle/ area of small circle
452.39/28.27 = 16.00
ratio is 1:16
change 4 5/9 from a mixed number to an improper fraction
Answers
Step-by-step explanation:
Hello, there!!
The answer would be 41/9.
The reason for above answer is to change any mixed fraction into improper fraction we should follow a simple step:
multiply the denominator with whole number.Add the answer (after mutiplied ).look here,
=[tex] \frac{4 \times 9 + 5}{9} [/tex]
we get 41/9.
Hope it helps...
The given fraction into the improper fraction should be [tex]\frac{41}{9}[/tex]
Given that,
The mixed number fraction is [tex]4 \frac{5}{9}[/tex]Computation:[tex]= 4\frac{5}{9}\\\\ = \frac{41}{9}[/tex]
Here we multiply the 9 with the 4 it gives 36 and then add 5 so that 41 arrives.
learn more about the fraction here: https://brainly.com/question/1301963?referrer=searchResults
A triangle has sides with lengths of 5x - 7, 3x -4 and 2x - 6. What is the perimeter of the triangle?
Answers
Answer:
Step-by-step explanation:
perimeter of triangle=sum of lengths of sides=5x-7+3x-4+2x-6=10x-17
Answer:
10x - 17
Step-by-step explanation:
To find the perimeter of a triangle, add up all three sides
( 5x-7) + ( 3x-4) + ( 2x-6)
Combine like terms
10x - 17
A researcher wants to determine the impact of soil type on the growth of a certain type of plant. She grows three plants in each of four different types of soil and measures the growth in inches for each plant after one month resulting in the data below.
Soil 1 Soil 2 Soil 3 Soil 4
12.6 12.2 12.2 11.1
12.6 12 10.6 11.7
14.3 13 9.1 9.6
1. What null hypothesis is the researcher testing if she runs an ANOVA with this data?
a.The mean growth of the plant in each type of soil is the same.
b. One type of soil has a higher mean growth for the plant than the others.
c. The variability in growth of the plant in each type of soil is the same.
d. Oil 3 provides a lower mean growth for the plant than the other types of soil.
e. The mean growth of the plant is different in each type of soil.
2. What is the SStrt for the ANOVA? Give your answer to at least three decimal places.
3. What is DFerr for the ANOVA?
4. What is the value of the F statistic for the ANOVA? Give your answer to at least three decimal places.
5. Using a 0.05 level of significance, what conclusion should the researcher reach?
a. There is not enough evidence to reject the claim that the mean growth of the plant is the same in each type of soil.
b. Soil 1 has a higher mean growth for the plant than the other types of soil.
c. The mean growth of the plant is not the same for all soil types .
d. Soil 3 has a lower mean growth for the plant than the other types of soil.
Answers
Answer:
(1) Option a
(2) 13.737
(3) 8
(4) 3.803
(5) Option a
Step-by-step explanation:
In this case, we need to determine whether the soil type effects the growth of a certain type of plant.
Perform the ANOVA test for the provided data on Excel.
Go to Data - Data Analysis - Anova: Single factor
Select the data for the growth.
Press OK.
The output is attached below.
(1)
The hypothesis for the study can be defined as follows:
H₀: The mean growth of the plant in each type of soil is the same.
Hₐ: The mean growth of the plant is different in each type of soil.
Correct option a.
(2)
The sum of square for treatment is:
[tex]\text{SS}_{trt}=\text{SS}_{BG}=13.737[/tex]
(3)
The degrees of freedom of error is:
[tex]\text{DF}_{err}=\text{DF}_{WG}=8[/tex]
(4)
The F statistic for the ANOVA is:
[tex]F=3.803[/tex]
(5)
The p-value of the test is:
[tex]p-value=0.058[/tex]
Decision Rule:
Reject H₀ if the p-value of the test is less than the level of significance.
[tex]\text{p-value}=0.058>\alpha=0.05[/tex]
The null hypothesis was failed to be rejected.
Conclusion:
There is not enough evidence to reject the claim that the mean growth of the plant is the same in each type of soil.
Correct option a.
The X- and y-coordinates of point P are each to be chosen at random from the set of integers 1 through 10.
What is the probability that P will be in quadrant II ?
О
1/10
1/4
1/2
Answers
Answer:
Ok, as i understand it:
for a point P = (x, y)
The values of x and y can be randomly chosen from the set {1, 2, ..., 10}
We want to find the probability that the point P lies on the second quadrant:
First, what type of points are located in the second quadrant?
We should have a value negative for x, and positive for y.
But in our set; {1, 2, ..., 10}, we have only positive values.
So x can not be negative, this means that the point can never be on the second quadrant.
So the probability is 0.
Which rule describes this transformation? (Zoom in to see it clearly)
Answers
Answer:
(x,y) -> (x+6, y-3)
Step-by-step explanation:
I followed c and it translated like the last ans choice.
Answer the question :)
Answers
Answer:
A. -11
Step-by-step explanation:
In the function, replace x with -2
R(x) = x^2 - 3x - 1 ➡ R(-2) = (-2)^2 - 3 × 2 -1 = -11
3 divided by 6 it hard
Answers
Answer:
3/6 = 1/2 = 0.5
Step-by-step explanation:
3 / 6 = 1/2 = 0.5
solve for x: 5x+3+8x-4=90
Answers
Answer:
[tex]x = 7[/tex]
Step-by-step explanation:
We can solve the equation [tex]5x+3+8x-4=90[/tex] by isolating the variable x on one side. To do this, we must simplify the equation.
[tex]5x+3+8x-4=90[/tex]
Combine like terms:
[tex]13x - 1 = 90[/tex]
Add 1 to both sides:
[tex]13x = 91[/tex]
Divide both sides by 13:
[tex]x = 7[/tex]
Hope this helped!
Answer:
x = 7
Step-by-step exxplanation:
5x + 3 + 8x - 4 = 90
5x + 8x = 90 - 3 + 4
13x = 91
x = 91/13
x = 7
probe:
5*7 + 3 + 8*7 - 4 = 90
35 + 3 + 56 - 4 = 90
Chloe wants to wrap a present in a box for Sarah. The top and bottom of the box is 8 in. by 3 in., the sides are both 3 in by 2 in. and the front and back are 8 in by 2 in. How much wrapping
paper will Chloe need to wrap the present?
Answers
Answer:
92 inches squared
Step-by-step explanation:
T/P = 8 * 3
L/R = 3 * 2
F/B = 8 * 2
Solving for surface area!
2(24) + 2(6) + 2(16) = 92
2. Use the diagram and given information to answer the questions and prove the statement.
a. Re-draw the diagram of the overlapping triangles so that the two triangles are separated.
b. What additional information would be necessary to prove that the two triangles, XBY and ZAY , are congruent? What congruency would be applied?
c. Prove (AZ) is congruent to (BX) using a flow chart proof. ( ):both have a line over them
Answers
[tex] \huge{ \underline{ \tt{ \purple{Solution:}}}}[/tex]
2) a)⚘ Refer to the attachment....
After separating, we will get two triangles △XYB and △ZYA where ∠Y is common to both the triangles, hence their measure is equal. This will be use in further proof.
b) We have,
∠X = ∠Z (Given, ATQ)∠Y = common to both triangles. XY = ZYSo, here
Two pairs of corresponding angles are equal along the side contained between them. So, The above triangles are congurent by ASA criterion.
✤ No more additional information Required to prove the above triangles be congurent.
➝ △XYB ≅ △ZYA (By ASA Criterion)
c) By using flow chart proof:
[tex] \boxed{ \sf{ \angle X = \angle Z}} \searrow[/tex]
[tex] \boxed{ \sf{\small{ \angle Y = com.}}} \rightarrow \boxed{\small{ \sf{ \triangle XYB \cong \triangle ZYA}}}\rightarrow \small{\boxed{ \sf{AZ= XB}}}[/tex]
[tex] \boxed{ \sf{XY = ZY}} \nearrow[/tex]
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Step-by-step explanation:
Hey mate ut answer is in the given attachment.
hope i help u
x/5=-2 . And how did you get it?
Answers
[tex]\dfrac{x}{5}=-2\\\\x=-10[/tex]
Answer:
[tex]\huge \boxed{{x=-10}}[/tex]
Step-by-step explanation:
[tex]\displaystyle \frac{x}{5} =-2[/tex]
We need the x variable to be isolated on one side of the equation, so we can find the value of x.
Multiply both sides of the equation by 5.
[tex]\displaystyle \frac{x}{5}(5) =-2(5)[/tex]
Simplify the equation.
[tex]x=-10[/tex]
The value of x that makes the equation true is -10.
The radius of a right circular cylinder is increasing at the rate of 7 in./sec, while the height is decreasing at the rate of 6 in./sec. At what rate is the volume of the cylinder changing when the radius is 20 in. and the height is 16 in.
Answers
Answer:
[tex]\approx \bold{6544\ in^3/sec}[/tex]
Step-by-step explanation:
Given:
Rate of change of radius of cylinder:
[tex]\dfrac{dr}{dt} = +7\ in/sec[/tex]
(This is increasing rate so positive)
Rate of change of height of cylinder:
[tex]\dfrac{dh}{dt} = -6\ in/sec[/tex]
(This is decreasing rate so negative)
To find:
Rate of change of volume when r = 20 inches and h = 16 inches.
Solution:
First of all, let us have a look at the formula for Volume:
[tex]V = \pi r^2h[/tex]
Differentiating it w.r.to 't':
[tex]\dfrac{dV}{dt} = \dfrac{d}{dt}(\pi r^2h)[/tex]
Let us have a look at the formula:
[tex]1.\ \dfrac{d}{dx} (C.f(x)) = C\dfrac{d(f(x))}{dx} \ \ \ (\text{C is a constant})\\2.\ \dfrac{d}{dx} (f(x).g(x)) = f(x)\dfrac{d}{dx} (g(x))+g(x)\dfrac{d}{dx} (f(x))[/tex]
[tex]3.\ \dfrac{dx^n}{dx} = nx^{n-1}[/tex]
Applying the two formula for the above differentiation:
[tex]\Rightarrow \dfrac{dV}{dt} = \pi\dfrac{d}{dt}( r^2h)\\\Rightarrow \dfrac{dV}{dt} = \pi h\dfrac{d }{dt}( r^2)+\pi r^2\dfrac{dh }{dt}\\\Rightarrow \dfrac{dV}{dt} = \pi h\times 2r \dfrac{dr }{dt}+\pi r^2\dfrac{dh }{dt}[/tex]
Now, putting the values:
[tex]\Rightarrow \dfrac{dV}{dt} = \pi \times 16\times 2\times 20 \times 7+\pi\times 20^2\times (-6)\\\Rightarrow \dfrac{dV}{dt} = 22 \times 16\times 2\times 20 +3.14\times 400\times (-6)\\\Rightarrow \dfrac{dV}{dt} = 14080 -7536\\\Rightarrow \dfrac{dV}{dt} \approx \bold{6544\ in^3/sec}[/tex]
So, the answer is: [tex]\approx \bold{6544\ in^3/sec}[/tex]
