Danny is measuring two pyramids whose bases are squares. $\newline$ Given the height $h$ and volume $V$ of the first pyramid, Danny uses the formula $\newline$ $a=\sqrt{\left(\frac{3V}{h}\right)}$ $\newline$ to compute its base's side length $a$ to be $5$ meters. $\newline$ The second pyramid has the same volume, but has $4$ times the height. What is the side length of its base?

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Q. Danny is measuring two pyramids whose bases are squares.

\newline

Given the height

h

and volume

V

of the first pyramid, Danny uses the formula

\newline

a=\sqrt{\left(\frac{3V}{h}\right)}

\newline

to compute its base's side length

a

to be

5

meters.

\newline

The second pyramid has the same volume, but has

4

times the height. What is the side length of its base?

Given Information: Given: $a = 5$ meters for the first pyramid, $V$ is the same for both pyramids, and the second pyramid's height is $4h$ .
Find Side Length of Second Pyramid: Use the formula $a = \sqrt{\left(\frac{3V}{h}\right)}$ to find the side length of the base of the second pyramid.
Substitute Values and Solve: Since the second pyramid's height is $4$ times the first, replace $h$ with $4h$ in the formula: $a = \sqrt{(\frac{3V}{4h})}.$
Isolate Volume $V$ : Substitute the known value of $a$ from the first pyramid into the formula to find $V$ : $5 = \sqrt{(\frac{3V}{h})}$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ . Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ . Divide both sides by $3$ to solve for $V$ : $V = \frac{25h}{3}$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ .Divide both sides by $3$ to solve for $V$ : $V = \frac{25h}{3}$ .Now, use the volume $V$ to find the side length $a$ of the second pyramid's base: $25 = \frac{3V}{h}$ $0$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ .Divide both sides by $3$ to solve for $V$ : $V = \frac{25h}{3}$ .Now, use the volume $V$ to find the side length $a$ of the second pyramid's base: $25 = \frac{3V}{h}$ $0$ .Substitute $V = \frac{25h}{3}$ into the formula: $25 = \frac{3V}{h}$ $2$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ .Divide both sides by $3$ to solve for $V$ : $V = \frac{25h}{3}$ .Now, use the volume $V$ to find the side length $a$ of the second pyramid's base: $25 = \frac{3V}{h}$ $0$ .Substitute $V = \frac{25h}{3}$ into the formula: $25 = \frac{3V}{h}$ $2$ .Simplify the formula: $25 = \frac{3V}{h}$ $3$ .
Calculate Side Length of Second Pyramid's Base: Square both sides to solve for $V$ : $25 = \frac{3V}{h}$ .Multiply both sides by $h$ to isolate $V$ : $25h = 3V$ .Divide both sides by $3$ to solve for $V$ : $V = \frac{25h}{3}$ .Now, use the volume $V$ to find the side length $a$ of the second pyramid's base: $25 = \frac{3V}{h}$ $0$ .Substitute $V = \frac{25h}{3}$ into the formula: $25 = \frac{3V}{h}$ $2$ .Simplify the formula: $25 = \frac{3V}{h}$ $3$ .Calculate the square root: $25 = \frac{3V}{h}$ $4$ .