you use a rectangular piece of cardboard 20 in by 30 to construct a box. you square the sides x in. from each corner of the piece of cardboard and then fold up the sides to form the bottom.
A. write a function A to represent the area of the base of the box in terms of x.
B. What is a reasonable domain for the function A?
C. Write an equation if the area of the base must be 416in^2
D. Solve the equation in part C for the values of x in the reasonable domain
What are the dimensions of the base of the box.
length of base = 30-2x
width = 20-2x
Area of base = (30-2x)(20-2x)
clearly 20-x and 30-2x have to be positve, or
0 < x < 10
(30-2x)(20-2x) = 416
600 - 100x + 2x^2 = 416
2x^2 - 100x + 184
x^2 - 50x + 92 = 0
I will complete the square
x^2 - 50x + 625 = -92 + 625
(x-25)^2 = 533
x-25 = ± √533
x = 25 ± √533
so staying within the domain x = 25+√533
or appr. 1.91
A. Let's start by visualizing the construction process. When we square the sides x inches from each corner and fold up the sides, we essentially create a rectangular box with a base that is 20-2x inches by 30-2x inches. Therefore, the function A, which represents the area of the base of the box, can be written as:
A(x) = (20 - 2x)(30 - 2x)
B. The reasonable domain for the function A would be when x satisfies the following conditions:
- x > 0: The squared sides cannot be negative.
- x < 10: The squared sides need to be less than half the length of the corresponding edge of the cardboard (otherwise, folding would be impossible).
- x < 15: The squared sides need to be less than half the width of the cardboard.
C. To find an equation for the area of the base, we can set A(x) equal to 416 in^2:
(20 - 2x)(30 - 2x) = 416
D. Now, let's solve the equation for the values of x in the reasonable domain:
(20 - 2x)(30 - 2x) = 416
600 - 60x - 40x + 4x^2 = 416
4x^2 - 100x + 184 = 0
Since this is a quadratic equation, we can solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case:
a = 4, b = -100, and c = 184.
Plugging these values into the quadratic formula and solving for x, we get:
x = (100 ± √(100^2 - 4 * 4 * 184)) / (2 * 4)
x = (100 ± √(10000 - 2944)) / 8
x = (100 ± √7056) / 8
x = (100 ± 84) / 8
Taking the positive value gives us:
x = (100 + 84) / 8
x = 20
Therefore, the value of x in the reasonable domain that satisfies the equation is x = 20.
To find the dimensions of the base of the box, we substitute x = 20 into the equation for A(x):
A(20) = (20 - 2(20))(30 - 2(20))
A(20) = 0(30 - 40)
A(20) = 0
This means that the dimensions of the base of the box are 0 inches by 0 inches. However, it's worth noting that the value x = 20 does not actually correspond to a valid solution in this case, as the fold would be impossible. Therefore, there are no valid dimensions for the base of the box given the constraints of the problem.
A. To represent the area of the base of the box in terms of x, we can subtract the areas of the four squares (each with side length x) from the total area of the cardboard, which is 20 in by 30 in.
Area of the base of the box = Area of the cardboard - 4 * Area of each square
Let A(x) represent the area of the base of the box in terms of x.
A(x) = (20 - 2x) * (30 - 2x)
B. The reasonable domain for the function A is the range of values for x that make sense in this context. Since we are squaring x and folding the sides up to form the bottom, x should be a value between 0 and half the length or width of the cardboard. In this case, the length and width of the cardboard are 20 in and 30 in, respectively.
So, a reasonable domain for the function A is 0 ≤ x ≤ 10.
C. The equation for the area of the base of the box, A(x), is given by:
A(x) = (20 - 2x) * (30 - 2x)
We need to solve the equation A(x) = 416 in^2.
(20 - 2x) * (30 - 2x) = 416
D. To solve the equation for the values of x in the reasonable domain (0 ≤ x ≤ 10), we can first simplify the equation:
(20 - 2x) * (30 - 2x) = 416
600 - 40x - 60x + 4x^2 = 416
4x^2 - 100x + 184 = 0
Now, we can either factor this quadratic equation or use the quadratic formula to solve for x.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a=4, b=-100, and c=184.
x = (-(-100) ± √((-100)^2 - 4(4)(184))) / 2(4)
x = (100 ± √(10000 - 2944)) / 8
x = (100 ± √7056) / 8
x = (100 ± 84) / 8
Now, we can find the values of x within the reasonable domain:
x = (100 + 84) / 8 = 184 / 8 = 23
x = (100 - 84) / 8 = 16 / 8 = 2
So, the dimensions of the base of the box are 23 in by 23 in or 2 in by 2 in.
A. To find the area of the base of the box in terms of x, we need to subtract the area of the four squares cut out from each corner. The dimensions of the base of the box will be reduced by 2x on each side (since you cut out x inches from each corner). Therefore, the length and width of the base will be 20 - 2x and 30 - 2x respectively. Thus, the function A can be represented as:
A(x) = (20 - 2x)(30 - 2x)
B. The domain of the function represents the valid values of x for which the formula A(x) is meaningful. In this case, since you are cutting squares out of each corner, the maximum value of x would be half the width or length of the cardboard. However, x cannot be greater than half the width or length because that would result in negative dimensions, which do not make sense. Therefore, a reasonable domain for the function A would be:
0 ≤ x ≤ min(20/2, 30/2) = 0 ≤ x ≤ 10
C. We are given that the area of the base must be 416in^2, so we can set up the following equation:
A(x) = (20 - 2x)(30 - 2x) = 416
D. To solve the equation A(x) = 416, we can expand the equation and bring all terms to one side to obtain a quadratic equation:
(20 - 2x)(30 - 2x) - 416 = 0
Expanding this expression, we get:
(600 - 80x - 60x + 4x^2) - 416 = 0
4x^2 - 140x + 184 = 0
We can then use factoring, completing the square, or using the quadratic formula to solve this equation. Once we find the values of x, we can substitute them back into the expressions for length and width to find the dimensions of the base of the box.