Question

type the correct answer in each box. use numerals instead of words. the length of a radius is three - fourths the height of a cone. the surface area is 3,750π square units. what are the height and the radius of the cone? the height is units. the radius is units.

Explanation:

Step1: Express radius in terms of height

Let the height of the cone be \(h\). Given \(r=\frac{3}{4}h\).
The slant height \(l=\sqrt{r^{2}+h^{2}}=\sqrt{(\frac{3}{4}h)^{2}+h^{2}}=\sqrt{\frac{9}{16}h^{2}+h^{2}}=\sqrt{\frac{9h^{2}+16h^{2}}{16}}=\sqrt{\frac{25h^{2}}{16}}=\frac{5}{4}h\).
The surface area of a cone \(A=\pi r(r + l)\).
Substitute \(r=\frac{3}{4}h\) and \(l=\frac{5}{4}h\) into the formula: \(A=\pi\times\frac{3}{4}h(\frac{3}{4}h+\frac{5}{4}h)\).

Step2: Simplify the surface - area formula and solve for \(h\)

Simplify \(A=\pi\times\frac{3}{4}h\times2h=\frac{3}{2}\pi h^{2}\).
Since \(A = 3750\pi\), we have \(\frac{3}{2}\pi h^{2}=3750\pi\).
Divide both sides by \(\pi\): \(\frac{3}{2}h^{2}=3750\).
Multiply both sides by \(\frac{2}{3}\): \(h^{2}=2500\).
Take the square root of both sides: \(h = 50\) (we take the positive value since \(h\) represents a length).

Step3: Find the radius

Since \(r=\frac{3}{4}h\), substitute \(h = 50\) into the formula. Then \(r=\frac{3}{4}\times50 = 37.5\).

Answer:

The height is \(50\) units. The radius is \(37.5\) units.