Question

determining the amount of metal for a trophy
there is 1,000 cm³ of aluminum available to cast a trophy that will be in the shape of a right square pyramid. is this enough aluminum for a trophy that has a base edge of 10 cm and a slant height of 13 cm?
○ no, the volume of the trophy is 1,200 cm³.
○ no, the volume of the trophy is 1,300 cm³.
○ yes, the volume of the trophy is 400 cm³.
○ yes, the volume of the trophy is 433.333 cm³

Explanation:

Step1: Find base half-edge length

Base edge = 10 cm, so half-edge = $\frac{10}{2}=5$ cm

Step2: Calculate pyramid height

Use Pythagorean theorem: $h=\sqrt{l^2 - s^2}$, where $l=13$ cm (slant height), $s=5$ cm.
$h=\sqrt{13^2 - 5^2}=\sqrt{169-25}=\sqrt{144}=12$ cm

Step3: Calculate base area

Base is square: $B = 10\times10=100$ cm²

Step4: Calculate pyramid volume

Volume formula: $V=\frac{1}{3}Bh$
$V=\frac{1}{3}\times100\times12=\frac{1200}{3}=400$ cm³

Step5: Compare to available aluminum

$400$ cm³ $< 1000$ cm³, so aluminum is enough.

Answer:

Yes, the volume of the trophy is 400 cm³.