Use similar triangles to find the area of a horizontal cross section at height y. Give your answer in terms of y.

Steps to Solve

1. Identify the relationship of similar triangles

In a cone, the similar triangles formed are between the triangle from the vertex to the base and the triangle from the vertex to the horizontal cross-section at height $y$. The full height of the cone is $h = 20$ and the base radius is $R = 4$.

2. Set up the proportions using similar triangles

Using the similar triangles, the ratio of the height to the radius must be the same. Thus, we can express this as:

$$ \frac{R}{h} = \frac{r}{y} $$

Where:

• $R$ is the radius of the base at height $h$ ($R = 4$)
• $r$ is the radius at height $y$

3. Rearranging the equation for $r$

Substituting the known values, we get:

$$ \frac{4}{20} = \frac{r}{y} $$

Cross-multiplying gives:

$$ 4y = 20r $$

Solving for $r$ yields:

$$ r = \frac{4y}{20} = \frac{y}{5} $$

4. Calculate the area of the cross-section

The area $A$ of the circular cross-section can be found using the formula for the area of a circle:

$$ A = \pi r^2 $$

Substituting $r = \frac{y}{5}$ into the area formula gives:

$$ A = \pi \left(\frac{y}{5}\right)^2 = \pi \frac{y^2}{25} $$