Free Printable Related Rate /calculus/

1.

1. Water is draining from the bottom of a cone-shaped funnel at the rate of $0.03\ \frac{ft^3}{\sec}$ . The height of the funnel is $2\ ft$ and the radius at the top of the funnel is $1\ ft$ . At what rate is the height of the water in the funnel changing when the height of the water is $12\ ft$ ?

a)

$\frac{0.48}{\pi}\ \frac{ft}{\sec}$

b)

$-0.153\ \frac{ft}{\sec}$

c)

$-4.8\ \frac{ft^2}{\sec}$

d)

$1.53\ \frac{ft}{\sec}$

2.

2. Car A is traveling west at $50\ \frac{mi}{h}$ and car B is traveling north at $60\ \frac{mi}{h}$ . Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is $0.3\ mi$ and car B is $0.4\ mi$ from the intersection?

a)

$-39\ \frac{mi}{h}$

b)

$-78\ \frac{mi}{h}$

c)

$78\ \frac{mi}{h}$

d)

$64\ \frac{mi}{h}$

3.

3. Air is being pumped into a spherical balloon so that its volume increases at a rate of $100\ \frac{cm^3}{\sec}$ . How fast is the radius of the balloon increasing when the diameter is $50\ cm$ ?

a)

$0.0127\ \frac{cm}{\sec}$

b)

$0.0127\ \ \frac{cm^2}{\sec}$

c)

$1.27\ \frac{cm}{\sec}$

d)

$0.00127\ \frac{cm}{\sec}$

4.

4. A rocket is launched so that it rises vertically. A camera is positioned $5000\ ft$ from the launch pad. When the rocket is $1000\ ft$ above the launch pad, its velocity is $600\ \frac{ft}{\sec}$ . Find the necessary rate of change of the camera’s angle as a function of time so that it stays focused on the rocket.

a)

$\frac{9}{26}\ \frac{ft}{\sec}$

b)

$\frac{9}{13}\ \frac{ft}{\sec}$

c)

$\frac{3}{13}\ \frac{ft}{\sec}$

d)

$\frac{3}{26}\ \frac{ft}{\sec}$

5.

5. A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of $0.5\ \frac{m^2}{\sec}$ at what rate is the radius decreasing when the area of the sheet is $12\ \frac{m}{\sec}$ ?

a)

$\frac{\sqrt{3}}{24\pi}\frac{cm}{\sec}$

b)

$\frac{-\sqrt{3}}{24\pi}\frac{cm}{\sec}$

c)

$\frac{-\sqrt{3\pi}}{24\pi}\frac{cm}{\sec}$

d)

$\frac{\sqrt{3\pi}}{24\pi}\frac{cm}{\sec}$

6.

6. A stone dropped in a pond sends out a circular ripple whose radius increases at a constant rate of $4\ \frac{ft}{\sec}$ . After 12 seconds, how rapidly is the area enclosed by the ripple increasing?

a)

$384\ \frac{ft^2}{\sec}$

b)

$832\ \frac{ft^2}{\sec}$

c)

$384\pi\ \frac{ft^2}{\sec}$

d)

$192\pi\ \frac{ft^2}{\sec}$

7.

7. A $15\ ft$ ladder is resting against the wall. The bottom is initially $10ft$ away from the wall and is being pushed towards the wall at a rate of $1.414\ \frac{ft}{\sec}$ . How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

a)

$-1319\ \frac{ft}{\sec}$

b)

$0.1319\ \frac{ft}{\sec}$

c)

$-0.1319\ \frac{ft}{\sec}$

d)

$1.319\ \frac{ft}{\sec}$

8.

8. A spotlight is on the ground $20ft$ away from a wall and a $6ft$ tall person is walking towards the wall at a rate of $2.5\ \frac{ft}{\sec}$ . How fast is the height of the shadow changing when the person is $8\ ft$ from the wall? Is the shadow increasing or decreasing in height at this time?

a)

$-\frac{25}{12}\ \frac{ft}{\sec}$

b)

$\frac{25}{12}\ \frac{ft}{\sec}$

c)

$2\ \frac{ft}{\sec}$

d)

$2.1\ \frac{ft}{\sec}$

9.

9. A conical paper cup is $10\ cm$ tall with a radius of $10\ cm.$ The cup is being filled with water so that the water level rises at a rate of $2\ \frac{cm}{\sec}$ . At what rate is water being poured into the cup when the water level is $8\ cm$ ?

a)

$127\pi\ \frac{cm^2}{\sec}$

b)

$128\pi\ \frac{cm^3}{\sec}$

c)

$128\pi\ \frac{cm^2}{\sec}$

d)

$127\pi\ \frac{cm^3}{\sec}$

10.

10. An airplane is flying overhead at a constant elevation of $4000\ ft$ . A man is viewing the plane from a position $3000\ ft$ from the base of radio tower. The airplane is flying horizontally from the man. If the plane is flying at the rate of $600\ \frac{ft}{\sec}$ at what rate is the distance between the man and the plane increasing when the plane passes over the radio tower?

a)

$150\ \frac{ft}{\sec}$

b)

$10\ \frac{ft}{\sec}$

c)

$1500\ \frac{ft}{\sec}$

d)

$0\ \frac{ft}{\sec}$

Related Rate /calculus/

Total questions: 10

Worksheet time: 46mins

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