Question

what would the value of x need to be to make the cone have a volume of 102π cm³? round to the nearest hundredth. cone diagram with height 8 cm and radius x

Explanation:

Step1: Recall cone volume formula

The volume \( V \) of a cone is given by \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Here, \( r = x \), \( h = 8 \) cm, and \( V = 102\pi \) \( \text{cm}^3 \).

Step2: Substitute values into formula

Substitute \( V = 102\pi \), \( h = 8 \) into \( V = \frac{1}{3}\pi r^2 h \):
\( 102\pi=\frac{1}{3}\pi x^2(8) \)

Step3: Solve for \( x^2 \)

Divide both sides by \( \pi \): \( 102=\frac{8}{3}x^2 \).
Multiply both sides by \( \frac{3}{8} \): \( x^2 = 102\times\frac{3}{8}=\frac{306}{8} = 38.25 \)

Step4: Solve for \( x \)

Take the square root of both sides: \( x=\sqrt{38.25}\approx 6.18 \) (rounded to nearest hundredth)

Answer:

\( 6.18 \)