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The radius of a sphere is increasing at a rate of 9 cm/sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

1 2 π cm

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The base of a triangle is shrinking at a rate of 1 cm/min and the height of the triangle is increasing at a rate of 5 cm/min. Find the rate at which the area of the triangle changes when the height is 22 cm and the base is 10 cm.

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A triangle has two constant sides of length 3 ft and 5 ft. The angle between these two sides is increasing at a rate of 0.1 rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is π / 6 .

The area is increasing at a rate ( 3 3 ) 8 ft 2 /sec.

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A triangle has a height that is increasing at a rate of 2 cm/sec and its area is increasing at a rate of 4 cm 2 /sec. Find the rate at which the base of the triangle is changing when the height of the triangle is 4 cm and the area is 20 cm 2 .

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For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

How fast does the depth of the water change when the water is 10 ft high if the cone leaks water at a rate of 10 ft 3 /min?

The depth of the water decreases at 128 125 π ft/min.

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Find the rate at which the surface area of the water changes when the water is 10 ft high if the cone leaks water at a rate of 10 ft 3 /min.

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If the water level is decreasing at a rate of 3 in./min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone.

The volume is decreasing at a rate of ( 25 π ) 16 ft 3 /min .

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A vertical cylinder is leaking water at a rate of 1 ft 3 /sec. If the cylinder has a height of 10 ft and a radius of 1 ft, at what rate is the height of the water changing when the height is 6 ft?

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A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of 2 m and a radius of 2 m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is 10 cm/min when the height is 1 m.

The water flows out at rate ( 2 π ) 5 m 3 /min.

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A trough has ends shaped like isosceles triangles, with width 3 m and height 4 m, and the trough is 10 m long. Water is being pumped into the trough at a rate of 5 m 3 /min . At what rate does the height of the water change when the water is 1 m deep?

A trough is shown with ends shaped like isosceles triangles. These triangles have width 3 and height 4. The trough is made up of rectangles that are of length 10. There is some water in the trough.
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A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of 2 3 m/sec?

An upside-down square pyramid is shown with square side lengths 4 and height 12. There is an unspecified amount of water inside the shape.

3 2 m/sec

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For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft 3 /min. The radius of the pool is 10 ft.

Find the rate at which the depth of the water is changing when the water has a depth of 5 ft.

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Find the rate at which the depth of the water is changing when the water has a depth of 1 ft.

25 19 π ft/min

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If the height is increasing at a rate of 1 in./sec when the depth of the water is 2 ft, find the rate at which water is being pumped in.

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Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 ft 3 /min. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 ft.

2 45 π ft/min

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Using a similar setup from the preceding problem, find the rate at which the gravel is being unloaded if the pile is 5 ft high and the height is increasing at a rate of 4 in./min.

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For the following exercises, draw the situations and solve the related-rate problems.

You are stationary on the ground and are watching a bird fly horizontally at a rate of 10 m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?

The angle decreases at 400 1681 rad/sec .

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You stand 40 ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/sec. Find the rate at which the angle of elevation changes when the rocket is 30 ft in the air.

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A lighthouse, L , is on an island 4 mi away from the closest point, P , on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?

A right triangle is formed by a lighthouse L, a point P on the shore that is perpendicular to the line from the lighthouse to the shore, and a point 2 miles to the right of the point P. The distance from P to L is 4 miles.

100 π / min

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Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach 1 mi away from the closest point on the beach.

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You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus travels west at a rate of 10 m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection?

The angle is changing at a rate of 21 25 rad/sec .

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For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.

A baseball field is shown, with the bases labeled Home, 1st, 2nd, and 3rd making a square with side lengths 90 ft.

[T] A batter hits a ball toward third base at 75 ft/sec and runs toward first base at a rate of 24 ft/sec. At what rate does the distance between the ball and the batter change when 2 sec have passed?

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[T] A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? ( Hint : Recall the law of cosines.)

The distance is increasing at a rate of 62.50 ft/sec.

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[T] A batter hits the ball and runs toward first base at a speed of 22 ft/sec. At what rate does the distance between the runner and second base change when the runner has run 30 ft?

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[T] Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of 18 ft/sec toward second base and the runner at second base runs at a speed of 20 ft/sec toward third base. How fast is the distance between runners changing 1 sec after the ball is hit?

The distance is decreasing at a rate of 11.99 ft/sec.

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Questions & Answers

find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
mujahid
How to do basic integrals
dondi Reply
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Jug
discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
ok ok hehe thanks nice dp ekko hahaha
Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
8x+12
Dhruv
8x+3
okhiria
d the derivative of y= e raised to power x
okhiria
Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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