Question

A homeowners' association limits the dimensions of the pools that it will allow in a particular subdivision. The bylaws state that permits will only be granted for pools shaped like rectangular prisms, for which the sum of the length of the pool and the perimeter of the vertical side containing the ladder cannot exceed 200 meters. The perimeter of the ladder side is determined using the width and the depth of the pool. If a pool has a length of 75 meters and its width is \(1.5\) times its depth, which of the following shows the allowable depth \(a\), in meters, of the pool? A) \(0

Short answer

The allowable depth a, in meters, of the pool is: \(0<a \leq 25\).

Step-by-step solution

1. Determine the sum of length (L) and the perimeter (P)

The length of the pool (L) is 75 meters. Let's denote the width of the pool as W and the depth as a. We are given that W = 1.5a. The perimeter of the vertical side containing the ladder equals the sum of the width and depth times 2: P = 2(W + a). We need to find the allowable depth (a) considering the restriction that: L + P ≤ 200. #Step 2: Replace the Width in the equation #

2. Replace the Width in the equation

We're given W = 1.5a. We can substitute this into the perimeter equation: P = 2(1.5a + a) #Step 3: Simplify the perimeter equation #

3. Simplify the perimeter equation

Simplify the equation: P = 2(1.5a + a) = 2(2.5a) = 5a #Step 4: Substitute the perimeter into the restriction equation #

4. Substitute the perimeter into the restriction equation

Now, we have to consider the restriction L + P ≤ 200. Substitute L (75) and P (5a) into the equation: 75 + 5a ≤ 200 #Step 5: Solve the inequality for a#

5. Solve the inequality for a

To find the allowable depth (a), we need to solve the inequality: 75 + 5a ≤ 200 Subtract 75 from both sides: 5a ≤ 125 Now, divide by 5: a ≤ 25 The allowable depth a, in meters, of the pool is: \(0<a \leq 25\). So, the correct answer is option D.

Key concepts

Understanding Rectangular Prisms

When we talk about rectangular prisms in the context of SAT math problems, we're referring to a three-dimensional solid object which has six faces that are rectangles. It's akin to a box, with a length, a width, and a height (or depth). Working with rectangular prisms involves knowing how to figure out various measurements like the volume, which is the amount of space inside the prism, and the surface area, which is the total area of all the external faces of the prism.

For problems like the one about pool dimensions governed by homeowners' association rules, recognizing that the structure is a rectangular prism helps to categorize and apply the correct formulas for perimeter or volume calculations. If the SAT problem mentions a rectangular prism, always be ready to use or manipulate these measurements with the formulas you've studied.

Perimeter Calculations in SAT Problems

Perimeter calculations are essential for many SAT math problems as they measure the boundary length of a shape. For two-dimensional shapes like rectangles or squares, you simply add up the lengths of all the sides. With three-dimensional objects like rectangular prisms, we often deal with the perimeter of one face of the object, especially when referring to a 'side containing a ladder,' as in our pool problem.

A critical aspect of perimeter calculations is understanding what dimensions to include. For a rectangular face, you add the lengths of all four sides, or simply calculate twice the sum of its length and width (or depth). This process was applied to find the perimeter of the pool's ladder side, which is a key step in determining the maximum allowable depth and ensuring the perimeter meets the homeowners' association's regulations.

Working with Inequalities

Inequalities are statements about the relative size or order of two values. They are fundamental in SAT math problems, often indicating a range of possible solutions or constraints on a measurement. In our example, the inequality tells us the combined length of the pool and the perimeter of one side must not exceed 200 meters. This is the kind of real-world scenario where inequalities are crucial in exemplifying restrictions and limitations.

Solving inequalities typically involves similar steps to solving equations: simplifying expressions, isolating the variable of interest, and performing arithmetic to find the range of permissible values. The algebraic manipulation tasks are the same, but the solution will often be a range or an interval as opposed to a single number. Understanding and correctly interpreting inequalities are important skills on the SAT, helping to answer questions that model real-life situations.