Math  /  Geometry

QuestionIf two pyramids are similar and the ratio between the lengths of their edges is 4:94: 9, what is the ratio of their volumes? A. 64:72964: 729 B. 81:1681: 16 C. 4:94: 9 D. 16:8116: 81

Studdy Solution

STEP 1

What is this asking? If we scale up a pyramid, how does the volume change? Watch out! Scaling factors apply differently to lengths, areas, and volumes!
Don't mix them up!

STEP 2

1. Relate the ratio of lengths to the ratio of volumes.

STEP 3

Alright, awesome students, let's tackle this pyramid problem!
We've got two **similar** pyramids, which means they have the same shape, but different sizes.
The problem tells us the ratio of their corresponding edge lengths is **4:9**.
We can think of this as scaling the smaller pyramid up to the size of the larger one.

STEP 4

Now, here's the key idea: when we scale a 3D shape, its volume changes by the **cube** of the scaling factor.
Why? Because volume is a 3D measure!
Think about a cube: if you double the length of its sides, you're not just doubling the volume, you're doubling it in *three* dimensions!

STEP 5

So, if the ratio of the edge lengths is 4:94:9, the scaling factor is 94\frac{9}{4}.
To find the ratio of their volumes, we need to **cube** this scaling factor.

STEP 6

Let's **calculate** that: (94)3=9343=999444=72964 \left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{9 \cdot 9 \cdot 9}{4 \cdot 4 \cdot 4} = \frac{729}{64} So, the ratio of the volumes is **729:64**.

STEP 7

The ratio of the volumes of the two pyramids is **729:64**, which corresponds to answer choice A.

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