Find the dimensions of an open topped box that can be made from 3600 cm^2 of cardboard, if the length of the base is three times as long as its width. What is the maximum volume?
a = 3w^2 + 2(wh + 3wh) = 3w^2+8wh
since a=3600,
h = (3600-3w^2)/8w
The volume is 3w^2h = 3w/8 (3600-3w^2)
find the derivative, set it to zero, and you will find that w=20.
So, h=15 and the maximum volume is 20x60x15 = 18000
To find the dimensions of the box, we need to use the given information and leverage the formulas for the area and volume of a rectangular box.
Let's start by representing the width of the base as "w." According to the question, the length of the base is three times as long as its width, so the length can be represented as "3w."
Since the box is open-topped, the area of the base will be equal to the area of the cardboard used. The area of the base is given by the formula: length × width.
Based on the given information, we can set up the equation: 3w × w = 3600 cm^2.
Next, let's solve this equation to find the value of "w," which represents the width of the base:
3w^2 = 3600
Divide both sides of the equation by 3:
w^2 = 1200
Take the square root of both sides:
w ≈ √1200
w ≈ 34.64 cm (rounded to two decimal places)
Now that we have the value of "w," we can find the length of the base by multiplying it by 3: length = 3w ≈ 3 × 34.64 ≈ 103.92 cm (rounded to two decimal places).
The dimensions of the base are approximately 34.64 cm (width) × 103.92 cm (length).
To find the maximum volume, we need to calculate the volume of the box using the formula: volume = length × width × height.
Since the box is open-topped, we can choose any height. Let's denote the height as "h."
Therefore, the maximum volume will be obtained by maximizing the height. To do this, we can set the height equal to the width, so h = w ≈ 34.64 cm.
Now, let's calculate the maximum volume:
volume = length × width × height
volume ≈ 103.92 cm × 34.64 cm × 34.64 cm
Using a calculator, we can find that the maximum volume is approximately 128,868.98 cm^3 (rounded to two decimal places).
Therefore, the dimensions of the open-topped box that can be made from 3600 cm^2 of cardboard are approximately:
- Width: 34.64 cm
- Length: 103.92 cm
- Height: 34.64 cm (to maximize the volume)
And the maximum volume is approximately 128,868.98 cm^3.