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Math  /  Geometry

QuestionThe two cones below are similar. What is the height of the larger cone? A. 5 B. 354\frac{35}{4} C. 285\frac{28}{5} D. 207\frac{20}{7}

Studdy Solution

STEP 1

What is this asking? We need to find the height of a bigger cone, knowing it's similar to a smaller cone with a height of 5 and base radius of 4, and that the bigger cone has a base radius of 7. Watch out! Similar shapes have proportional sides, not equal sides!
Don't mix up the ratios.

STEP 2

1. Set up the proportion
2. Solve for *x*

STEP 3

Since the cones are similar, the ratio of corresponding sides must be equal.
This means the ratio of the height of the larger cone to the height of the smaller cone is equal to the ratio of the radius of the larger cone to the radius of the smaller cone.
Let's write that down!

STEP 4

We can write this relationship as: height of larger coneheight of smaller cone=radius of larger coneradius of smaller cone \frac{\text{height of larger cone}}{\text{height of smaller cone}} = \frac{\text{radius of larger cone}}{\text{radius of smaller cone}} Substituting the given values, we get: x5=74 \frac{x}{5} = \frac{7}{4}

STEP 5

To solve for xx, we want to get it by itself.
We can do this by multiplying both sides of the equation by **5**.
Remember, what we do to one side, we *must* do to the other!
This keeps the equation balanced.
Think of it like a seesaw!

STEP 6

x5⋅5=74⋅5 \frac{x}{5} \cdot 5 = \frac{7}{4} \cdot 5 On the left side, the **5** in the numerator and denominator divide to one, leaving just xx.
Awesome!

STEP 7

x=7⋅54 x = \frac{7 \cdot 5}{4} x=354 x = \frac{35}{4} So, the height of the larger cone is 354\frac{35}{4}!

STEP 8

The height of the larger cone is 354\frac{35}{4}, so the answer is B.

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