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-length

The base edge is 10 cm, so half the edge is $\frac{10}{2}=5$ cm.

Step2: Calculate pyramid height

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

Step3: Calculate base area

Base is square: $A = 10\times10=100\text{ cm}^2$

Step4: Calculate pyramid volume

Volume formula: $V=\frac{1}{3}Ah$
$$V=\frac{1}{3}\times100\times12=400\text{ cm}^3$$

Step5: Compare to available aluminum

$400\text{ cm}^3 < 1000\text{ cm}^3$, so aluminum is enough.

Answer:

Yes, the volume of the trophy is 400 cm³.