Mathematics for Management -- Supplementary Electronic Materials

Related Rates

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1. The radius of a circle is changing at the rate of $\frac{1}{\pi}$ centimeters per second. At what rate, in square centimeters per second, is the circle's area changing when $r = 5$ cm?

$ \frac{5}{\pi} $ $ 10 $ $ \frac{10}{\pi} $ $ 15 $

2. The volume of a cube is increasing at the rate of $12$ cm$^3$/min . How fast is the surface area increasing, in square centimeters per minute, when the length of an edge is $20$ cm?

$ 1 $ $ \frac{6}{5} $ $ \frac{4}{3} $ $ \frac{12}{5} $

3. When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, then the radius is

$ \frac{1}{4 \pi} $ $ \frac{1}{4} $ $ \frac{1}{\pi} $ $ 1 $

4. The radius $r$ of a sphere is increasing at a constant rate. At the time when the surface area $S = 4 \pi r^2$ and the radius of the sphere are increasing at the same numerical rate, what is the radius of the sphere?

$ \frac{1}{8 \pi} $ $ \frac{1}{4 \pi} $ $ \frac{1}{3 \pi} $ $ \frac{\pi}{8} $

5. The radius $r$ of a sphere is increasing at the rate of $0.3$ centimeters per second. At the instant when the surface area $S = 4 \pi r^2$ becomes $100 \pi$ cm$^2$, what is the rate of increase, in cm$$^3$/s, in the volume $V = \frac{4}{3} \pi r^3$.

$ 10 \pi $ $ 12 \pi $ $ 25 \pi $ $ 30 \pi $

6. If the radius $r$ of a cone is decreasing at a rate of $2$ centimeters per minute while its height $h$ is increasing at a rate of $4$ centimeters per minute, which of the following must be true about the volume $V$ of the cone? ($V = \frac{1}{3} \pi r^2 h$)

$ V $ is always decreasing $ V $ is always increasing $ V$ is increasing only when $r > h $ $ V$ is increasing only when $r < h $

7. If the base $b$ of a triangle is increasing at a rate of $3$ centimeters per minute while its height $h$ is decreasing at a rate of $3$ centimeters per minute, which of the following must be true about the area $A$ of the triangle?

$ A$ is always increasing $ A$ is always decreasing $ A$ is decreasing only when $b < h $ $ A$ is decreasing only when $b > h $

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

$ -\frac{18}{5} $ $ -18 $ $ -\frac{17}{5}$ $ -3 $

9. When $x = 8$, the rate at which $\sqrt[3]{x}$ is increasing is $\frac{1}{k}$ times the rate at which $x$ is increasing. What is the value of $k$?

$ 3 $ $ 4 $ $8 $ $ 12 $

10. Let $f(x) = \sqrt{x}$. Assume that the rate of $f(x)$ at $x = c$ is twice the rate of change at $x = 1$ then what is the value of $c$?

$ \frac{1}{4} $ $ \frac{1}{\sqrt{2}} $ $ 4 $ $ \frac{1}{2 \sqrt{2}} $

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Solution: 1b; 2d; 3c; 4a; 5d; 6c; 7d; 8a; 9d; 10c

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.

Related Rates

1. The radius of a circle is changing at the rate of \(\frac{1}{\pi}\) centimeters per second. At what rate, in square centimeters per second, is the circle's area changing when \(r = 5\) cm?

2. The volume of a cube is increasing at the rate of \(12\) cm\(^3\)/min . How fast is the surface area increasing, in square centimeters per minute, when the length of an edge is \(20\) cm?

3. When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, then the radius is

4. The radius \(r\) of a sphere is increasing at a constant rate. At the time when the surface area \(S = 4 \pi r^2\) and the radius of the sphere are increasing at the same numerical rate, what is the radius of the sphere?

5. The radius \(r\) of a sphere is increasing at the rate of \(0.3\) centimeters per second. At the instant when the surface area \(S = 4 \pi r^2\) becomes \(100 \pi\) cm\(^2\), what is the rate of increase, in cm$\(^3\)/s, in the volume \(V = \frac{4}{3} \pi r^3\).

6. If the radius \(r\) of a cone is decreasing at a rate of \(2\) centimeters per minute while its height \(h\) is increasing at a rate of \(4\) centimeters per minute, which of the following must be true about the volume \(V\) of the cone? (\(V = \frac{1}{3} \pi r^2 h\))

7. If the base \(b\) of a triangle is increasing at a rate of \(3\) centimeters per minute while its height \(h\) is decreasing at a rate of \(3\) centimeters per minute, which of the following must be true about the area \(A\) of the triangle?

8. Water slowly evaporates from a circular shaped puddle. The area of the puddle decreases at a rate of \(36 \pi\) cm\(^2\)/h. Assuming the puddle retains its circular shape, at what rate, in cm/h, is the radius of the puddle changing when the radius is \(5\) cm?

9. When \(x = 8\), the rate at which \(\sqrt[3]{x}\) is increasing is \(\frac{1}{k}\) times the rate at which \(x\) is increasing. What is the value of \(k\)?

10. Let \(f(x) = \sqrt{x}\). Assume that the rate of \(f(x)\) at \(x = c\) is twice the rate of change at \(x = 1\) then what is the value of \(c\)?