Question

a tetrahedron has a vertical height of 8.5 cm and a volume of 105 cm³. find the base area of the tetrahedron (correct to the nearest tenth of a cm²)

Explanation:

Step1: Recall the volume formula for a tetrahedron

The volume \( V \) of a tetrahedron is given by the formula \( V=\frac{1}{3}Bh \), where \( B \) is the base area and \( h \) is the height (altitude) of the tetrahedron.

Step2: Substitute the given values into the formula

We are given that \( V = 100\space cm^{3} \) and \( h=8.5\space cm \). Substituting these values into the formula \( V=\frac{1}{3}Bh \), we get:
\[
100=\frac{1}{3}\times B\times8.5
\]

Step3: Solve for the base area \( B \)

First, multiply both sides of the equation by 3 to get rid of the fraction:
\[
3\times100 = B\times8.5
\]
\[
300=8.5B
\]
Then, divide both sides by \( 8.5 \) to solve for \( B \):
\[
B=\frac{300}{8.5}=\frac{3000}{85}=\frac{600}{17}\approx35.29\space cm^{2}
\]

Answer:

The base area of the tetrahedron is \(\frac{600}{17}\space cm^{2}\) (or approximately \( 35.29\space cm^{2} \))