We know that this increase is 120cm², so we can write:
2x(w + 2h) + 4h² - 72w - 72h = 120
Simplifying this equation, we get:
x(w + 2h) + 2h² - 36w - 36h = 60
Since we want to find the volume of the box, we need to use the formula:
V = lwh
We know that the original length is 36cm, so we can write:
V = 36wh
We also know that the new length is x, so we can write:
V' = xwh
The increase in volume is:
V' - V = xwh - 36wh = (x - 36)wh
We know that the increase in surface area is 120cm², so we can write:
2x(w + 2h) + 4h² - 72w - 72h = 120
Dividing both sides by 2, we get:
x(w + 2h) + 2h² - 36w - 36h = 60
Rearranging this equation, we get:
x(w + 2h) - 36w = 36h - 2h² + 60
Substituting w = 36/h - 2h/3 into this equation, we get:
x(36/h + 4h/3) - 36(36/h - 2h/3) = 36h - 2h² + 60
Multiplying both sides by 3h, we get:
108x + 144h² - 3888 + 288h² = 108h - 6h³ + 180h
Simplifying this equation, we get:
6h³ - 240h² + 108h + 3888 - 108x = 0
Dividing both sides by 6, we get:
h³ - 40h² + 18h + 648 - 18x = 0
We can use this equation to solve for h, and then we can use h to solve for x and the volume of the box. However, this equation does not have a nice solution, so we will need to use numerical methods to approximate the solution.
Answers & Comments
Answer:
h³ - 40h² + 18h + 648 - 18x = 0
Step-by-step explanation:
Let x be the length of each part after cutting.
The original surface area of the box is:
2lw + 2lh + 2wh = 2(36)(w + h) + 2wh = 72w + 72h + 2wh
After cutting, the new surface area is:
2xw + 2xh + 2wh + 4xh = 2x(w + h) + 2h(x + 2h) + 2wh = 2xw + 2xh + 2wh + 2xh + 4h²
= 2x(w + 2h) + 2wh + 4h²
The increase in surface area is:
2x(w + 2h) + 2wh + 4h² - (72w + 72h + 2wh) = 2x(w + 2h) + 4h² - 72w - 72h
We know that this increase is 120cm², so we can write:
2x(w + 2h) + 4h² - 72w - 72h = 120
Simplifying this equation, we get:
x(w + 2h) + 2h² - 36w - 36h = 60
Since we want to find the volume of the box, we need to use the formula:
V = lwh
We know that the original length is 36cm, so we can write:
V = 36wh
We also know that the new length is x, so we can write:
V' = xwh
The increase in volume is:
V' - V = xwh - 36wh = (x - 36)wh
We know that the increase in surface area is 120cm², so we can write:
2x(w + 2h) + 4h² - 72w - 72h = 120
Dividing both sides by 2, we get:
x(w + 2h) + 2h² - 36w - 36h = 60
Rearranging this equation, we get:
x(w + 2h) - 36w = 36h - 2h² + 60
Substituting w = 36/h - 2h/3 into this equation, we get:
x(36/h + 4h/3) - 36(36/h - 2h/3) = 36h - 2h² + 60
Multiplying both sides by 3h, we get:
108x + 144h² - 3888 + 288h² = 108h - 6h³ + 180h
Simplifying this equation, we get:
6h³ - 240h² + 108h + 3888 - 108x = 0
Dividing both sides by 6, we get:
h³ - 40h² + 18h + 648 - 18x = 0
We can use this equation to solve for h, and then we can use h to solve for x and the volume of the box. However, this equation does not have a nice solution, so we will need to use numerical methods to approximate the solution.