91Ó°ÊÓ

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.

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.

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.

A radius of circle is$r$. The circumference of the circle is $2\mathrm{Ï€°ù}.$

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.

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

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.

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

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.

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

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

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

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

Thus, the statement is True.

The circumference of the circle of the cylinder is given by $2\mathrm{Ï€°ù}$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 $2\mathrm{Ï€°ù}$and length $h.$

Thus, the statement is True.

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.