a piece of rectangular sheet metal is 20 in. wide. It is made into a rain gutter by turning up the edges to form parallel sides. Let x represent the length of each of the parallel side.
(a)give the restrictions on x
(b)determine a fuction A that gives the area of a cross section of the gutter.
(c) For what value of x will A be a maximum (and thus minimize the amount of water that the gutter will hold)? what is this maximum area?
(d) for what values of x will the area of a cross section be less than 10 in.^2?
See related questions below.
(a) In order to form a rain gutter, the length of each parallel side (x) should be smaller than the width of the rectangular sheet metal. Therefore, the restriction on x is: x < 20 inches.
(b) To determine the area of a cross-section of the gutter, we need to subtract the areas of the rectangular sheet metal that are turned up to form the parallel sides.
Since the width of the rectangular sheet metal is 20 inches, and x represents the length of each parallel side, the length of the rectangular sheet metal remaining in the center can be represented as (20 - 2x) inches (subtracting twice the length of each parallel side from the width).
The cross-sectional shape of the gutter is a rectangle. Therefore, the area of the cross-section (A) can be calculated using the formula:
A = (20 - 2x) * x
(c) To find the value of x that will maximize the area (A) and minimize the amount of water the gutter can hold, we need to find the vertex of the quadratic function A = (20 - 2x) * x.
The vertex of a quadratic function with the form A = ax^2 + bx + c can be found using the formula: x = -b / (2a)
In our case, a = -2 and b = 20. Plugging these values into the formula, we get:
x = -20 / (2*-2)
x = -20 / -4
x = 5
Therefore, the value of x that will maximize the area is 5 inches.
To find the corresponding maximum area, we substitute this value back into the area function:
A = (20 - 2x) * x
A = (20 - 2*5) * 5
A = (20 - 10) * 5
A = 10 * 5
A = 50 square inches
So, when x = 5 inches, the maximum area of the cross section is 50 square inches.
(d) We want to find the values of x for which the area of a cross-section is less than 10 in^2.
We can set up an inequality using the area function:
A < 10
Substituting the area function with (20 - 2x) * x, we get:
(20 - 2x) * x < 10
Expanding the equation, we have:
20x - 2x^2 < 10
Rearranging the equation to have zero on one side, we have:
2x^2 - 20x + 10 > 0
Now, we can solve this quadratic inequality. However, to simplify the solution, we can divide the entire inequality by 2. This gives us:
x^2 - 10x + 5 > 0
The solutions to this inequality will be the values of x for which the area of the cross-section is less than 10 in^2.
Using factoring, the quadratic equation can be factored into:
(x - 5)(x - 1) > 0
By observing the signs, we can determine the intervals where the inequality holds true. In this case, the inequality holds true when either:
(x - 5) > 0 and (x - 1) > 0
or
(x - 5) < 0 and (x - 1) < 0
Solving these inequalities, we find that x > 5 or x < 1.
Therefore, the values of x for which the area of a cross-section is less than 10 in^2 are x < 1 inches and x > 5 inches.