Question

a recycling bin is in the shape of a rectangular box. find the height of the box if its length is 19 ft, its width is 10 ft, and its surface area is 786 ft². (in the figure, h = height. assume that the given surface area includes that of the top lid of the box.) the height of the box is \boxed{} ft.

Explanation:

Step1: Recall surface area formula for rectangular box

The surface area \( S \) of a rectangular box with length \( l \), width \( w \), and height \( h \) is given by the formula:

$$ S = 2(lw + lh + wh) $$

We know \( l = 19 \) ft, \( w = 10 \) ft, and \( S = 786 \) \( \text{ft}^2 \). Substitute these values into the formula:

$$ 786 = 2(19 \times 10 + 19h + 10h) $$

Step2: Simplify the equation

First, calculate \( 19 \times 10 = 190 \). Then combine like terms inside the parentheses: \( 19h + 10h = 29h \). So the equation becomes:

$$ 786 = 2(190 + 29h) $$

Divide both sides by 2:

$$ \frac{786}{2} = 190 + 29h $$

$$ 393 = 190 + 29h $$

Step3: Solve for \( h \)

Subtract 190 from both sides:

$$ 393 - 190 = 29h $$

$$ 203 = 29h $$

Divide both sides by 29:

$$ h = \frac{203}{29} = 7 $$

Answer:

The height of the box is \( \boxed{7} \) ft.