Q. 1 - Calculus

Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: If a square grows larger, so that its side length increases at a constant rate, then its area will also increase at a constant rate.

(b) True or False: If a square grows larger, so that its side length increases at a constant rate, then its perimeter will also increase at a constant rate. 

(c) True or False: If a circle grows larger, so that its radius increases at a constant rate, then its circumference will also increase at a constant rate.

(d) True or False: If a sphere grows larger, so that its radius increases at a constant rate, then its volume will also increase at a constant rate.

(e) True or False: The volume of a right circular cone is one-third of the volume of the right circular cylinder with the same radius and height.

(f) True or False: If $V\left(r\right)$ is the volume of a sphere as a function of its radius, and $S\left(r\right)$ is the surface area of a sphere as a function of its radius, then$V\left(r\right)=S\left(r\right)$.

(g) True or False: If you unroll the side of a right circular cylinder with radius $r$ and height $h$, you get a flat rectangle with height $h$ and width $2\mathrm{\pi r}.$

(h) True or False: Given a right triangle with side lengths $a$ and $c$ and hypotenuse of length $b$, we must have$a^2+b^2=c^2.$ 

  

Step-by-Step Solution

Answer

Part (a) The statement is True.

Part (b) The statement is True. 

Part (c) The statement is True. 

Part (d) The statement is True. 

Part (e) The statement is True. 

Part (f) The statement is True. 

Part (g) The statement is True. 

Part (h) The statement is False. 

Part (a) Step 1. Given information

If a square grows larger, so that it’s side length increases at a constant rate, then its area will also increase at a constant rate.

Part (a) Step 2. Explanation

A side length of square is $L.$ The area of the square is $L^2.$

If the square grows larger , then the rate of change in the area $\frac{dA}{dt}$ depends on both the side length and the area.

This means when the side length increases at a constant rate, then its area will also increase at a constant rate.

Thus, the statement is true.

Part (b) Step 1 Explanation

A side length of square is $L$. The perimeter of the square is $4L.$

If the square grows larger , then the rate of change in the perimeter $\frac{dP}{dt}$ depends on both the side length and the perimeter.

This means when the side length increases at a constant rate, then its perimeter will also increase at a constant rate.

Thus, the statement is true.

Part (c) Step 1. Explanation

A radius of circle is $r$. The circumference of the circle is $2\mathrm{\pi r}.$

If the circle grows larger, then the rate of change in the circumference $\frac{dC}{dt}$ depends on both the radius and the circumference.

This means when the radius increases at a constant rate, then its circumference will also increase at a constant rate.

Thus, the statement is true.

Part (d) Step 1. Explanation

A radius of sphere is $r$. The volume of the sphere is $V=\frac{4}{3}{\mathrm{\pi r}}^3.$

If the sphere grows larger, then the rate of change in the volume $\frac{dV}{dt}$  depends on both the radius and the volume.

This means when the radius increases at a constant rate, then its volume will also increase at a constant rate.

Thus, the statement is true.

Part (e) Step 1. Explanation

A volume of the right circular cone is $V_2=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ and a volume of the right circular cylinder is $V_1={\mathrm{\pi r}}^2\mathrm{h}$.

So, $V_2=\frac{1}{3}{V}_1$

This means volume of the right circular cone is one-third of the volume of a right circular cylinder.

Thus, the statement is True.

Part (f) Step 1. Explanation

The volume of a sphere as a function of its radius is given by $V\left(r\right)=\frac{4}{3}{\mathrm{\pi r}}^3.$

So the first derivative is given by $V\text{'}\left(r\right)=4{\mathrm{\pi r}}^2$.

The surface area of a sphere as a function of its radius is given by $S\left(r\right)=4{\mathrm{\pi r}}^2.$

This means $V\text{'}\left(r\right)=S\left(r\right)$

Thus, the statement is True.

Part (g) Step 1. Explanation

The circumference of the circle of the cylinder is given by $2\mathrm{\pi r}$ is the width of the unrolled cylinder.

The height of the cylinder $h$ is the length of the unrolled cylinder.

This means the unrolled cylinder is a rectangle with width $2\mathrm{\pi r}$ and length $h.$

Thus, the statement is True.

Part (h) Step 1. Explanation

The Pythagorean theorem states that if a right triangle has legs of lengths $a$ and $b$ and hypotenuse $c$. Then,

$a^2+b^2=c^2$

Thus, the statement is False.