I'm really bad at story problems in math so can someone explain how to do this probablom to me?

Question

This is my last homework problem and I just need help with part a after I get part a I'll be able to do b-d. Here it is:

An open box with a square base is required to have a volume of 10 cubic feet.

a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base.

p.s. A = Area

And can you explain the steps you used to get the answer. I don't want you to just do the problem for me I wanna know how to do it myself. Thanx

ya bandf like that really answers the question. I already drew the stupid box so I have it visualized but I don't know how to find the answer.

No further answered needed bandf came back and explained the question really really well. He will get the 10 points no matter what other answers I get. Thanx for all you help:)

Answer 1

You have to first visualize how the box (an open box with no lid) will be made. My assumption is that you are required to have a base (area = x * x) and then four sides (area = x * height). Does that make sense to you so far?

The volume of this box will be x * x * h and this must equal 10 cubic feet. So h = 10 / x²

So let's put it all together. The material used to make the box will be:

The base: x²
The four sides: 4 * x * (10 / x²)

This last part can be simplifed by canceling an x:

The base: x²
The four sides: 4 * (10 / x) = 40 / x

Add these together to get a function for the total area of material:
f(x) = x² + (40 / x)

Let's check this answer out, by thinking of a box with a base of 1 ft. square. To make 10 cubic feet, this would be 10 feet high. So the four sides would be 1 x 10 and the base would be 1 x 1. This is a total of 41 sq. ft. Check that with the formula:
f(1) = 1² + (40 / 1)
f(1) = 1 + 40
f(1) = 41 sq. ft.

Try another example, with x = 2. Here the base will be 4 sq. ft and the height must be 2.5 ft to make it 10 cu. ft. So 2 x 2 for the base and 2 x 2.5 for the four sides. That's 4 + 4 * 5 = 24 sq. ft.

f(2) = 2² + 40/2
f(2) = 4 + 20
f(2) = 24 sq. ft.

So this formula seems to accurately express the amount of material used to make a box given the length x for the base.
f(x) = x² + (40 / x)

Note: the next question will probably be, what value of x will minimize the amount of material used. To determine that, either graph the equation and find the minimum point, or take the derivative. In this case:
f'(x) = 2x - 40 / x²

Set this to 0:
2x - 40 / x² = 0

Add 40 / x² to both sides:
2x = 40 / x²

Multiply by x²:
2x³ = 40
x³ = 20
x = ³√20

So the amount of material will be minimized when x is the cube root of 20 (about 2.7144 ft.) I know this is more than you asked, but I just wanted to figure it out for myself, since I was curious.

Answer 2

The amount of material is the combined surface area of all 6 sides....this is not the same as the volume, which must be 10 cubic feet.

So, the three dimensions of the box are x, x, and y (you know both bottom sides are the same length, call it x). Now you know that x * x * y = 10 (the required volume).

The surface area is x * x (the bottom) plus x * x (the top) plus x * y * 4 (the 4 other sides).

Now you have 2 equations, you can solve for the required surface area in terms of x.

Does this help?

Answer 3

Looking at total surface area without a lid to the box. (We'll also assume that there are no flaps on the box's bottom, because that would require extra material.)

The area of the bottom is x^2.
The height we'll call h.
Now (x^2)h = 10 since that is the formula for the volume
h = 10/(x^2) solving for h in terms of x

Area of bottom is x^2.
Area of each side is xh.
There are 4 sides so the toal area of the 4 sides is 4xh.
Add up the 4 sides with the base: 4xh + x^2
Substitute 10/(x^2) for h to get
4x/(x^2) + x^2
Now simplify
(4/x) + x^2 is the formula for the total amount of material needed for this box.

Answer 4

Volume=10cft
volume=s^3
s^3=10
s=(10)^1/3
area of the six faces A=6a^2
A=6[(10)^1/3]^2
A=6*(10)^2/3 sft

Answer 5

ok, so the fee of extra valuable fulfillment is, i think, the utmost proportion made. on account that the two women have made 21 photos, you are able to placed it into basic fractions like so: amy- 8/21, and jane- 4/21. now amy has the extra valuable fulfillment fee using fact in case you divide 8 by making use of 21 you will get .38095. as a proportion that is composed of 38% mutually as doing a similar with janes score 4 divided by making use of 21 it provide you .19047= 19%.