4. A trapezoidal gutter is to be made, from a strip of metal 24 inches wide by bending up the edges at a certain angle. What width across the top gives the greatest carrying capacity? A. 16 inches B. 15 inches C. 12 inches D. 10 inchesSee answer

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4 A trapezoidal gutter is to be made from a strip of metal 24 inches wide by bending up the edges at a certain angle What width across the top gives the greatest carrying capacity A 16 inches B 15…

Question

4. A trapezoidal gutter is to be made, from a strip of metal 24 inches wide by bending up the edges at a certain angle. What width across the top gives the greatest carrying capacity?
A. 16 inches
B. 15 inches
C. 12 inches
D. 10 inches

Basic Answer

Correct Answer:

C. 12 inches

Analyzing the Answer：

To determine the width across the top that gives the greatest carrying capacity for a trapezoidal gutter, we need to consider the geometry and the area of the trapezoid. The strip of metal is 24 inches wide, and bending up the edges forms the sides of the trapezoid.

Let’s denote:

• $b$ as the width across the top (the smaller base of the trapezoid),
• $B$ as the width across the bottom (the larger base of the trapezoid, which is 24 inches),
• $h$ as the height of the trapezoid.

The area $A$ of a trapezoid is given by:
$$A=\frac{1}{2}\times (B+b)\times h$$

To maximize the carrying capacity, we need to maximize the area $A$. The height $h$ is determined by the angle at which the edges are bent up. For simplicity, let’s assume the height $h$ is such that the trapezoid is symmetric about its center line.

Given that the total width of the metal strip is 24 inches, the sum of the lengths of the two slanted sides (which are equal) and the width across the top $b$ must equal 24 inches. Therefore:
$$2\times \text{length of slanted side}+b=24$$

To maximize the area, the width across the top $b$ should be half of the width across the bottom $B$. This is because the area of the trapezoid is maximized when the trapezoid is symmetric and the slanted sides are equal. Thus:
$$b=\frac{B}{2}=\frac{24}{2}=12\text{ inches}$$

Analysis of other options：

A：The width across the top being 16 inches is incorrect because it does not maximize the area of the trapezoid.
B：The width across the top being 15 inches is incorrect for the same reason as above.
D：The width across the top being 10 inches is incorrect because it does not maximize the area of the trapezoid.