Question

the height of a cone is four - thirds times the radius. the lateral area is 135π square units. what is the slant height of the cone? type the correct answer in each box. use numerals instead of words. for help, see this worked example. the slant height is units.

Explanation:

Step1: Recall the lateral - area formula

The lateral - area formula of a cone is $A = \pi rl$, where $r$ is the radius and $l$ is the slant height. We are given that $A = 135\pi$. So, $\pi rl=135\pi$. Divide both sides by $\pi$ to get $rl = 135$.

Step2: Express height in terms of radius and find slant - height in terms of radius

We know that the height $h=\frac{4}{3}r$. By the Pythagorean theorem for a cone, $l=\sqrt{h^{2}+r^{2}}$. Substitute $h = \frac{4}{3}r$ into the formula for $l$: $l=\sqrt{(\frac{4}{3}r)^{2}+r^{2}}=\sqrt{\frac{16}{9}r^{2}+r^{2}}=\sqrt{\frac{16r^{2}+9r^{2}}{9}}=\sqrt{\frac{25r^{2}}{9}}=\frac{5}{3}r$.

Step3: Solve for $r$

Since $rl = 135$ and $l=\frac{5}{3}r$, substitute $l$ into $rl = 135$: $r\times\frac{5}{3}r = 135$. This simplifies to $\frac{5}{3}r^{2}=135$. Multiply both sides by $\frac{3}{5}$: $r^{2}=135\times\frac{3}{5}=81$. Take the square - root of both sides: $r = 9$ (we consider the positive value since radius is a non - negative quantity).

Step4: Solve for $l$

Since $l=\frac{5}{3}r$ and $r = 9$, then $l=\frac{5}{3}\times9 = 15$.

Answer:

15