Calculus AB

Related Rates

100

Suppose water is being poured into a cubical tank at a rate of 100 ft³/min. The side of the cube measures 20ft. Find the rate at which the depth of the water is changing with respect to time.

.25 ft/min

200

A boat is pulled in by means of a winch on the dock 12 ft. above the deck of the boat. The winch pulls in rope at a rate of 4 ft/sec. Determine the speed of the boat when there is 13 ft. of rope out.

10.4 ft/sec

300

The height of a rectangular box is 10 in. Its length increases at a rate of 2 in/sec while its width decreases at a rate of 4 in/sec. When the length is 8 in. and the width is 6 in., find the rate at which the volume of the box is changing.

-200 in³/sec

400

A 6 ft. tall man is walking towards a building at a rate of 5 ft/sec. Suppose there is a light 50 ft. from the building, determine how fast the man's shadow on the building is shortening when he is 30 ft. from the building.

-3.75 ft/sec

500

A hemispherical water tank has a radius of 6 meters and is losing water. The area of the surface of the water is A=12πh-πh² square meters, where h is the depth of the water in the tank. When h=3 meters, the depth of the water is decreasing at a rate of .5 meters per minute. Find the rate at which the area of the water's surface is decreasing with respect to time at that instant.

3π

Area

100

Find the area between the curve y=x and x=4 in the first quadrant.

8

200

Find the area under the curve y=x² from x=2 to x=3.

19

___

3

300

Find the area under the curve y=e^2x from x=1 to x=4.

e⁸ - e²

_____

2

400

Find the area under the curve y=6^2x from x=0 to x=2.

1295

_____

ln36

500

Find the area under the curve y=tan(x) from x=π/4 to x=π.

-ln(2)^1/2

Volume

100

Find the volume of the solid formed by rotating about the x-axis the region enclosed by the graphs of y=x²+1, the x and y axes, and the line x=2.

43.145

200

Find the volume of the solid formed by rotating about the y-axis the region enclosed by the graphs of y=x^1/2+1, the y-axis, and the line y=4.

152.681

300

Find the volume of the solid formed by rotating the region enclosed by y=x^1/2 and y=x² about the line x=-2.

5.131

400

The base of a solid is the region enclosed by y=e^x, the x and y axes, and the line x=ln(2). The cross sections perpendicular to the x-axis are squares. Find the volume of the solid.

1.5

500

The base of a solid is the region enclosed by y=x^1/2 and y=x². The cross sections perpendicular to the y-axis are equilateral triangles. Find the volume of the solid.

.056

Derivative

100

Find the derivative of x⁷+7.

7x⁶

200

Find the derivative of csc(2x).

-2csc(2x)cot(2x)

300

Find the derivative of y=6^log₆^tan(9x).

9sec²(9x)

400

Find the derivative of y=(xsinx)^1/2.

xcosx+sinx

_________

2(xsinx)^1/2

500

Derivative of csc^-1(x⁵+2x).

−5x⁴-2
___________________

|x⁵+2x|[(x⁵+2x)²-1]^1/2

Pos, Vel, Acc, Spe

100

What is the velocity of a particle when s(t)=3t² at t=3 (m/sec)

18 m/sec

200

What is the acceleration of a particle with the position given as s(t)=4t³-7t?

24t

300

What is the total distance traveled by a particle if s(t)=t³-3t²-4t at t=5? (in meters)

approximately 56.257 meters

400

A particle moves along the x-axis with velocity given by v(t)=3t² -4 for time t>=0. If the particle is at position x=-2 at time t=0, what is the position of the particle at time of the particle at time t = 3

13

500

If v(t)=t²+6t-9, a(4)=14, s(0)=3, and speed is increasing at t=6, then what is the total distance traveled at t=4?

45.157