two cones are similar. the surface area of the larger cone is 65π square inches. the surface area of the…

Explanation:

Step1: Recall the ratio - of - areas for similar solids

For two similar solids, the ratio of their surface areas is equal to the square of the ratio of their corresponding linear dimensions (such as radius). Let the radius of the larger cone be $R$ and the radius of the smaller cone be $r = 6.4$ inches. Let the surface area of the larger cone be $A_1=65\pi$ and the surface area of the smaller cone be $A_2 = 41.6\pi$. Then $\frac{A_1}{A_2}=\left(\frac{R}{r}\right)^2$.

Step2: Substitute the given values into the formula

Substitute $A_1 = 65\pi$ and $A_2=41.6\pi$ into the formula $\frac{A_1}{A_2}=\left(\frac{R}{r}\right)^2$. First, simplify $\frac{A_1}{A_2}$: $\frac{65\pi}{41.6\pi}=\frac{65}{41.6}=\frac{650}{416}=\frac{25}{16}$. So, $\frac{25}{16}=\left(\frac{R}{6.4}\right)^2$.

Step3: Solve for $R$

Take the square - root of both sides of the equation $\frac{25}{16}=\left(\frac{R}{6.4}\right)^2$. We get $\frac{R}{6.4}=\sqrt{\frac{25}{16}}=\frac{5}{4}$. Cross - multiply to solve for $R$: $R=\frac{5}{4}\times6.4$. Since $\frac{5}{4}\times6.4 = 5\times1.6=8$ inches.

Answer:

A. 8 inches