thank you in advance
An x by x square is cut from each corner of a rectangular sheet of cardboard that is 13 inches wide and 17 inches long. The sides are then folded up to form a box.
1.Write an equation to represent the volume of the box.
2.To the nearest tenth, what value of x maximizes the volume of the box?
hint: volume = length x width x height
1. To write an equation to represent the volume of the box, we need to understand the dimensions of the box after folding.
If an x by x square is cut from each corner of the rectangular sheet of cardboard, the resulting dimensions of the box will be:
Width: 13 - 2x inches
Length: 17 - 2x inches
Height: x inches
The volume of a rectangular box is calculated by multiplying its width, length, and height. Therefore, the equation for the volume, V, of the box can be written as:
V = (13 - 2x) * (17 - 2x) * x
2. To find the value of x that maximizes the volume of the box, we can differentiate the volume equation with respect to x and set the derivative equal to zero. Then, we solve for x.
dV/dx = (17 - 2x)(13 - 2x) + x(13 - 2x) * (-4) + x(17 - 2x) * (-4) = 0
Simplifying the equation and solving for x requires some algebraic manipulation. Let's proceed step by step:
1st term:
(17 - 2x)(13 - 2x) = 221 - 34x - 26x + 4x^2 = 4x^2 - 60x + 221
2nd term:
x(13 - 2x) * (-4) = -52x + 8x^2
3rd term:
x(17 - 2x) * (-4) = -68x + 8x^2
Combining all terms:
4x^2 - 60x + 221 - 52x + 8x^2 - 68x + 8x^2 = 0
Combine like terms:
20x^2 - 180x + 221 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 20, b = -180, and c = 221.
Plugging in the values into the quadratic formula, we get:
x = (-(-180) ± sqrt((-180)^2 - 4 * 20 * 221)) / (2 * 20)
x = (180 ± sqrt(32400 - 17680)) / 40
x = (180 ± sqrt(14720)) / 40
Calculating the square root of 14720 gives us:
x = (180 ± 121.29) / 40
x ≈ (180 + 121.29) / 40 ≈ 5.515
Since x represents the length of a side, rounding it to the nearest tenth gives us:
x ≈ 5.5
Therefore, to the nearest tenth, the value of x that maximizes the volume of the box is approximately 5.5 inches.
To answer these questions, let's go step by step:
1. To write the equation representing the volume of the box, we need to find the dimensions of the box first. After cutting squares of side length x from each corner of the rectangular sheet, the dimensions of the resulting box would be (13-2x) inches (length), (17-2x) inches (width), and x inches (height).
The volume of the box is given by the formula V = length × width × height. So, substituting in the values, the equation representing the volume of the box is:
V = (13 - 2x) × (17 - 2x) × x
2. To find the value of x that maximizes the volume of the box, we need to determine the maximum value by maximizing the equation for the volume of the box.
To do this, we can take the derivative of the volume equation with respect to x and set it equal to zero. Then solve for x to find the critical point.
Let's differentiate the volume equation:
dV/dx = [13 - 2x) × (17 - 2x)]' × x + (13 - 2x) × [(17 - 2x) × x]' + x × [(13 - 2x) × (17 - 2x)]'
= [(13 - 2x) × (17 - 2x)]' × x + (13 - 2x) × (17 - 2x) - x × (13 - 2x) × (17 - 2x)'
where ' denotes differentiation.
Simplifying further, we have:
0 = [(13 - 2x) × (17 - 2x)]' × x + (13 - 2x) × (17 - 2x) - x × (13 - 2x) × (17 - 2x)'
Now, solve this equation for x. Since it involves higher-level math concepts, it might be more suitable to solve it using a graphing calculator or computational software. By finding the critical point, you can determine the value of x that maximizes the volume of the box.
Remember to round the answer to the nearest tenth as specified in the question.
Once you find the value of x, you can substitute it back into the volume equation to calculate the corresponding maximum volume of the box.