A company is selling 1 gallon blocks of orange juice. your task is to design a rectangle box ( prism) that meet all the criteria below:
a) contains at least 231 cubic inches.
b)the height of the box must be exactly 1/4th of the width of the box.
c)the box is closed on all 6 sides.
d) the surface area of the box minimized.?
let the height be x
then the width is 4x , (the height is 1/4 of the width, this way I have no fractions )
let the length be y
volume = (x)(4x)(y) = 4x^2 y
4x^2 y = 231
y = 231/(4x^2)
SA = 2(4xy) +2(x)(4x) + 2(xy)
= 8xy + 8x^2 + 2xy
= 10xy+8x^2
= 10x(231/x^2) + 8x^2
= 2310/x + 8x^2
d(SA)/dx = -2310/x^2 + 16x
= 0 for a min of SA
16x = 2310/x^2
x^2 = 2310/16 = 1155/8
x = (1155)^(1/3)/2 = appr 5.25 inches
4x = 21.0 inches
y = 2.10 inches
The box should be 5.25 inches by 21 inches by 2.1 inches
check:
for my answer:
5.25 x 21 x 2.1 = 231.55
SA = 660.5
if x = 5
Volume = 231
SA = 662 which is > 660.5
if x = 4
volume = 231
SA = 705.5 which is > 660.5
My answer is correct
Hmm. I got
v = 5.29 x 13.22 x 3.30 = 230.78
a = 262.12
yup, looked over my work and found an error in
my SA substitution.
SA = 2(4xy) +2(x)(4x) + 2(xy)
....
= 10x(231/x^2) + 8x^2 ---->10x(231/(4x^2)) + 8x^2
to produce your answers
To design a rectangle box (prism) that meets all the given criteria, we need to consider the requirements regarding the volume, height, closed sides, and minimized surface area. Let's break it down step by step:
a) Volume: The box should contain at least 231 cubic inches. The formula to calculate the volume of a rectangular box is V = length * width * height.
Let's assume the width of the box is represented by 'w'. According to the given criteria, the height of the box should be exactly 1/4th of the width. Thus, the height is 'w/4'.
Now, we need to calculate the length 'l' of the box. We can use the formula for volume:
231 = l * w * (w/4)
231 = (l * w^2)/4
To simplify this equation, we can multiply both sides by 4:
924 = l * w^2
So, we have an equation 924 = l * w^2, which satisfies the desired volume requirement.
b) Height: The height of the box should be exactly 1/4th of the width.
c) Closed Sides: A rectangular box has 6 sides - 4 side faces, 1 top face, and 1 bottom face. To meet the criteria, all 6 sides should be closed.
d) Minimized Surface Area: The surface area of a rectangular box can be calculated using the formula SA = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
To minimize the surface area, we need to minimize the sum of each side's area. From the given criteria, we know that the height will be 1/4th of the width. We can substitute this value into the surface area formula to get a simplified equation in terms of a single variable (width).
SA = 2lw + 2l(w/4) + 2w(w/4)
SA = 2lw + (l/2)w + (w^2)/2
Now, we can substitute the value of length 'l' from the equation (924 = l * w^2) we found earlier:
SA = 2(924/w^2)w + (924/w^2)w/2 + (w^2)/2
SA = 1848/w + 462/w + (w^2)/2
To find the value of 'w' that minimizes the surface area, we can take the derivative of the surface area function with respect to 'w' and set it equal to zero. Then we can solve the resulting equation to find the value of 'w' that minimizes the surface area.
After calculating the derivative, we get:
d(SA)/dw = -1848/(w^2) + (462 - w^2)/(2w^2)
Setting this derivative equal to zero and solving for 'w', we can find the width that minimizes the surface area.