A rectangular box has a square base with side length x inches and a height of 5 inches. Th… — SAT

A rectangular box has a square base with side length $x$ inches and a height of $5$ inches. The total surface area of the box is $192$ square inches. What is the value of $x$ ?

Answer: 6 inches

Answer

The value of

x

6

The total surface area is found by adding the areas of all six faces. The top and bottom faces are squares with side length

x

, so their combined area is

2x^2

. The four vertical sides are rectangles with dimensions

x

5

, so their combined area is

4(5x) = 20x

. Setting their sum equal to the total surface area gives

2x^2 + 20x = 192

. Dividing by

2

yields

x^2 + 10x - 96 = 0

, which factors as

(x + 16)(x - 6) = 0

. Since

x

must be positive,

x = 6

Step-by-Step Solution

Set up the equation for the total surface area of the rectangular box.

\text{Total Surface Area} = 2x^2 + 20x = 192

The total surface area of a rectangular box with a square base of side length

x

and height

h

consists of two square bases (top and bottom) of area

x^2

each, and four rectangular sides of area

xh

each. Substituting

h = 5

gives

2x^2 + 4(5x) = 2x^2 + 20x

Simplify and set the quadratic equation to zero.

x^2 + 10x - 96 = 0

Dividing all terms by

2

simplifies the equation to

x^2 + 10x = 96

. Subtracting

96

from both sides sets the quadratic equation to standard form

ax^2 + bx + c = 0

Factor the quadratic equation and solve for the positive value of

x

x = 6

Factoring the equation gives

(x + 16)(x - 6) = 0

, which yields solutions of

x = -16

and

x = 6

. Since the side length of a geometric solid must be positive, we reject the negative solution.

Key Concept

Calculating the surface area of a rectangular prism and solving the resulting quadratic relationship.

Estimated Time:1m 35s

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