Question

yuma and his brother are painting a sandbox shaped like a rectangular prism. they want to paint only the outsides of the four rectangular sides. how many square feet will be painted? (not drawn to scale) image of a rectangular prism sandbox with dimensions 12 ft (length), 10 ft (width), and 1 1/4 ft (height) a 55 ft² b 60 ft² c 50 ft² d 65 ft²

Explanation:

Step1: Identify the sides to paint

The sandbox is a rectangular prism, and we paint the four lateral sides (excluding top and bottom). There are two pairs of rectangular sides: one pair with dimensions \(12\) ft (length) and \(1\frac{1}{4}\) ft (height), and another pair with dimensions \(10\) ft (width) and \(1\frac{1}{4}\) ft (height).

Step2: Calculate area of one pair (length - height)

First, convert \(1\frac{1}{4}\) to an improper fraction: \(1\frac{1}{4}=\frac{5}{4}\) ft.
The area of one rectangle with length \(12\) ft and height \(\frac{5}{4}\) ft is \(12\times\frac{5}{4} = 15\) square feet.
Since there are two such rectangles, their total area is \(2\times15 = 30\) square feet.

Step3: Calculate area of the other pair (width - height)

The area of one rectangle with width \(10\) ft and height \(\frac{5}{4}\) ft is \(10\times\frac{5}{4}=\frac{50}{4} = 12.5\) square feet.
For two such rectangles, the total area is \(2\times12.5 = 25\) square feet.

Step4: Total area to paint

Add the areas of the two pairs: \(30 + 25 = 55\) square feet. Wait, no—wait, let's recheck. Wait, \(12\times\frac{5}{4}=15\), two of those: \(30\). \(10\times\frac{5}{4}=12.5\), two of those: \(25\). \(30 + 25 = 55\)? But wait, maybe I made a mistake. Wait, no—wait, the height is \(1\frac{1}{4}\), length \(12\), width \(10\). The lateral surface area (four sides) of a rectangular prism is \(2\times(\text{length}\times\text{height}+\text{width}\times\text{height})\). So that's \(2\times(12\times1\frac{1}{4}+10\times1\frac{1}{4})\). Factor out \(1\frac{1}{4}\): \(2\times1\frac{1}{4}\times(12 + 10)\). \(1\frac{1}{4}=\frac{5}{4}\), so \(2\times\frac{5}{4}\times22=\frac{5}{2}\times22 = 55\). Wait, but the options have 55 as option A? Wait, but let's check again. Wait, maybe I miscalculated. Wait, \(12\times1.25 = 15\), two of them: 30. \(10\times1.25 = 12.5\), two of them: 25. 30 + 25 = 55. So the total area is 55 square feet. Wait, but the options: A is 55, B is 60, C is 50, D is 65. So the correct answer should be A? Wait, but wait—wait, maybe the height is the depth? Wait, no, the problem says "the outsides of the four rectangular sides". So the lateral surface area. Wait, maybe I messed up the formula. Wait, lateral surface area of a rectangular prism is \(2h(l + w)\), where \(h\) is height, \(l\) length, \(w\) width. So \(h = 1\frac{1}{4}=\frac{5}{4}\), \(l = 12\), \(w = 10\). So \(2\times\frac{5}{4}\times(12 + 10)= \frac{5}{2}\times22 = 55\). Yes, that's correct. So the total area is 55 square feet. Wait, but earlier I thought maybe I made a mistake, but no. So the answer is 55, which is option A. Wait, but let me check again. Wait, \(12\times1.25 = 15\), two sides: 30. \(10\times1.25 = 12.5\), two sides: 25. 30 + 25 = 55. Yes. So the correct answer is A.

Answer:

A. \(55\ \text{ft}^2\)