# Radius of a Cylinder

## Task

Given the height $h$ and volume $V$ of a certain cylinder, Jill uses the formula $$r=\sqrt{\frac{V}{\pi h}}$$ to compute its radius to be 20 meters. If a second cylinder has the same volume as the first, but is 100 times taller, what is its radius?

- $2$ meters.
- $200$ meters.
- $0.2$ meters.
- $2,000$ meters.
- It is impossible to tell from the given information.

## IM Commentary

This task is part of a joint project between Student Achievement Partners and Illustrative Mathematics to develop prototype machine-scorable assessment items that test a range of mathematical knowledge and skills described in the CCSSM and begin to signal the focus and coherence of the standards.

### Task Purpose

The purpose of this task is to assess

A-SSE.1: Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret $P(1+r)^n$ as the product of $P$ and a factor not depending on $P$.

In the task, students must reason directly from the structure of the expression on the right hand side of the formula, because they are not given the initial volume and height and so do not have enough information to recalculate the radius directly. They must realize that multiplying $h$ by 100 has the effect of multiplying the expression by $1/10$, for example by reasoning directly from the position of $h$ in the denominator of a fraction inside a square root, or by writing $$ r=\sqrt{\frac{V}{\pi h}}=\left(\sqrt{\frac{V}{\pi}}\right)\cdot h^{-1/2}. $$

### Cognitive Complexity

##### Mathematical Content

The task is about seeing structure in expressions, as explained above.

##### Mathematical Practices

The task addresses MP7, Look for and make use of structure, which is a thread through the entire A-SSE domain.

##### Linguistic Demand

The volume context requires moderate reading comprehension.

##### Stimulus Material

There is a verbal description of the context of the task and an equation that involves a square root of a rational expression which is moderately complex.

##### Response Mode

Multiple choice is not complex.

## Solution

We are given that with the initial values of $V$ and $h$, the radius calculation gave

$$ r=20=\sqrt{\frac{V}{\pi h}}. $$ We now observe that increasing $h$ by a factor of 100 has the effect of decreasing the expression $\sqrt{\frac{V}{\pi h}}$ by a factor of 10. Indeed, $$ \sqrt{\frac{V}{\pi (100h)}}=\sqrt{\frac{1}{100}\frac{V}{\pi h}}=\frac{1}{\sqrt{100}}\sqrt{\frac{V}{\pi h}}=\frac{1}{10}\sqrt{\frac{V}{\pi h}}=\frac{1}{10}\cdot 20=2. $$ That is, the radius of the second cylinder is one tenth the radius of the first cylinder, giving a radius of 2 meters.

## Radius of a Cylinder

Given the height $h$ and volume $V$ of a certain cylinder, Jill uses the formula $$r=\sqrt{\frac{V}{\pi h}}$$ to compute its radius to be 20 meters. If a second cylinder has the same volume as the first, but is 100 times taller, what is its radius?

- $2$ meters.
- $200$ meters.
- $0.2$ meters.
- $2,000$ meters.
- It is impossible to tell from the given information.