A rectangular box has Dimensions 16 inches and 22 inches and volume 4224 cubic inches. If the same amount is decreased from the largest and smallest sides, the volume of the new box is 6/11 of the first box. Find the dimensions of the second box.
original :
16 by 22 by x
352x = 4224
x = 12
So the first box is 22 by 16 by 12
let the amount decreased by y , clearly y < 12
then (22-y)(16)(12-y) = (6/11)(4224)
16(352 - 34y + y^2) = 2304
y^2 - 34y + 352 = 144
y^2 - 34y + 208 = 0
(y-26)(y-8) = 0 , but y < 12
so y = 8
so the second box is 14 by 16 by 4
To find the dimensions of the second box, we need to first calculate the new dimensions of the first box after decreasing the largest and smallest sides by the same amount.
Let's assume the amount decreased from both sides is 'x' inches.
The original dimensions of the box are:
Length = 16 inches
Width = 22 inches
So, after decreasing the largest and smallest sides by 'x' inches, the new dimensions become:
Length = (16 - x) inches
Width = (22 - x) inches
Now, let's calculate the volume of the first box using the original dimensions:
Volume = Length * Width * Height
Given, Volume = 4224 cubic inches
So, 4224 = 16 * 22 * Height
Dividing both sides by (16 * 22):
Height = 4224 / (16 * 22)
Height = 12 inches (approximately)
Now, let's calculate the volume of the new box (second box):
Volume = Length * Width * Height
Given, volume of the new box = 6/11 * volume of the first box
So, volume of the new box = (6/11) * 4224
volume of the new box = 2304 cubic inches
We need to find the dimensions of the second box, which means we need to find the new Length and Width.
Using the formula for the volume of the new box, we have:
2304 = (16 - x) * (22 - x) * 12
Simplifying this equation will yield a quadratic equation, which can be solved to find the value of 'x'.
The quadratic equation is:
x^2 - 38x + 192 = 0
Solving this equation using the quadratic formula will give us two possible values for 'x'. We can then substitute these values into the equations:
Length = (16 - x) inches
Width = (22 - x) inches
to find the corresponding dimensions of the second box.