Question 1

A square has a side length that is increasing at a rate of $$2\ \mathrm{in/sec}. If $$the original side length is $$8\ \mathrm{in}$$, what is the rate of change of the area of the square at that instant?

Accepted Answer

$$32\ \mathrm{in}^2/\mathrm{sec}$$

Question 2

Given $$A = x^2$ for the area of a square, and $$\frac{dx}{dt} = 2$$, x = 8$, what is $$\frac{dA}{dt}$$?

Accepted Answer

$$\frac{dA}{dt} = 2x \frac{dx}{dt} = 2(8)(2) = 32$$

Question 3

If the side length of a square is increasing at a constant rate, how does the rate of change of the area depend on the side length?

Accepted Answer

It is directly proportional to the side length.

Question 4

Suppose the side length of a square is $$x$$ inches and is increasing at a rate of $$k$$ inches per second. Which formula gives the rate of change of the area?

Accepted Answer

$$\frac{dA}{dt} = 2xk$$

Question 5

A square's side length is increasing at $$2\ \mathrm{in/sec}. $$After $$3$$ seconds, what is the rate of change of the area, assuming the original side length was $$8\ \mathrm{in}$$?

Accepted Answer

$$40\ \mathrm{in}^2/\mathrm{sec}$$

Question 6

If the side length of a square is $$x$$ and $$\frac{dx}{dt} is $$negative, what does this imply about $$\frac{dA}{dt}$$?

Accepted Answer

$$\frac{dA}{dt} is $$negative, so the area is decreasing.

Question 7

A square's area is increasing at $$32\ \mathrm{in}^2/\mathrm{sec}. If $$the side length is $$8\ \mathrm{in}$$, what is the rate of change of the side length?

Accepted Answer

$$2\ \mathrm{in/sec}$$

Question 8

If the side length of a square doubles, how does the rate of change of the area (with respect to time) change, assuming $$\frac{dx}{dt} is $$constant?

Accepted Answer

It doubles.

Question 9

Which of the following best describes the relationship between the rate of change of the area and the rate of change of the side length for a square?

Accepted Answer

The rate of change of the area is twice the side length times the rate of change of the side length.

Question 10

A square metal plate is being heated so that each side expands at $$2\ \mathrm{in/sec}. If $$the original side is $$8\ \mathrm{in}$$, how fast is the area increasing after $$5$$ seconds?

Accepted Answer

$$48\ \mathrm{in}^2/\mathrm{sec}$$

Question 11

If the area of a square is increasing at a constant rate, what can you say about the rate of change of the side length?

Accepted Answer

It decreases as the side length increases.

Question 12

Suppose the side length of a square is $$x$$ inches and is increasing at $$k$$ inches per second. If $$x$$ triples, what happens to $$\frac{dA}{dt}$$?

Accepted Answer

It triples.

Related Rates: Rate of Change of Area of a Square

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