Calculus Related Rates Practice

When working with related rates problems, we must first find a formula to relate known quantities and unknown quantities. Then, we must ​ (a)   the formula with respect to ​ (b)   .

• differentiate
• time
• integrate
• distance
• multiply
• cheese
• add
• apples

To solve a related rates problem, we must use a known ​ (a)   , to solve for an unknown quantity.

• rate of change
• constant value
• static measurement
• fixed quantity

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What equation(s) should be used?

What is the derivative of circumference with respect to time?

What is the derivative of area with respect to time?

Water slowly evaporates from a circular shaped puddle. The area of the puddle decreases at a rate of $36\pi\ \frac{in^2}{hr}$ . Assuming the puddle retains its circular shape, at what rate is the radius of the puddle changing when the radius is 5 in?

•  $\frac{-\frac{18}{5}in}{hr}$  

•  $\frac{-\frac{17}{5}in}{hr}$  

•  $-18\ \frac{in}{hr}$  

•  $-3\ \frac{in}{hr}$  

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 3 cm/min. How fast is the area of the pool increasing when the radius is 12 cm?

• $76\pi\ \frac{cm^2}{\min}$

• $72\pi\frac{cm^2}{\min}$

• $77\pi\frac{cm^2}{\min}$

• $62\pi\frac{cm^2}{\min}$

A hypothetical cube grows at a rate of  $64\ \frac{m^3}{\min}$ . How fast are the sides of the cube increasing when the sides are 8 m?

•  $\frac{\frac{2}{3}m}{\min}$  

•  $\frac{1m}{\min}$  

•  $\frac{\frac{1}{3}m}{\min}$  

•  $\frac{\frac{7}{3}m}{\min}$  

A spherical snowball is rolled in fresh snow, causing it to grow so that its radius increases at a rate of 2 in/sec. How fast is the volume of the snowball increasing when the radius is 2 in?

• $42\pi\ \frac{in^3}{\sec}$

• $39\pi\frac{in^3}{\sec}$

• $22\pi\ \frac{in^3}{\sec}$

• $32\pi\frac{in^3}{\sec}$

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the top of the ladder is 5 ft high?

• 3 ft/sec

• 7.2 ft/sec

• -12 ft/sec

• -7.2 ft/sec