a box has a surface area of 184 sq in give the dimensions it could have for its length width and height hint find three areas that add up to 92 square inches
we know that 2lw + 2lh + 2wh = 184
or
their hint: lw + lh + wh = 92
there would be an infinite number of solutions,
e.g.
let w = 4, l = 8, then
4(8) + 8h + 4h = 92
12h = 60
h = 5
so length = 8, width = 4, and height = 5
check: 4*8 + 4*5 + 8*5 = 92
let w = 1, h = 6
l + 6l + 6 = 92
7l = 86
l = 86/7
so length = 86/7 , width = 1, height = 6
check: (86/7)(1) + (86/7)(6) + 6 = 92
etc
To find the dimensions of the box with a surface area of 184 sq in, we need to find three areas that add up to 92 sq in. Let's assume the dimensions of the box are length (L), width (W), and height (H).
1. The first area is the product of the length and width, which is L * W.
2. The second area is the product of the length and height, which is L * H.
3. The third area is the product of the width and height, which is W * H.
We can set up the equation:
L * W + L * H + W * H = 92
Now, let's try different dimensions to satisfy this equation and find the possible values.
1. Let's assume one side of the box has a length of 10 inches.
L = 10
Replacing the value of L in the equation:
10 * W + 10 * H + W * H = 92
2. Next, let's assume the width is 8 inches:
W = 8
Replacing the value of W in the equation:
10 * 8 + 10 * H + 8 * H = 92
80 + 10H + 8H = 92
18H = 12
H = 12 / 18
H = 0.67 inches
Therefore, the possible dimensions for the box with a surface area of 184 sq in are as follows:
Length (L) = 10 inches
Width (W) = 8 inches
Height (H) = 0.67 inches.
To find the dimensions of a box given its surface area, we need to consider that a box has six faces. By understanding this, we can break down the surface area into three pairs of faces.
Let's denote the three dimensions of the box as length (L), width (W), and height (H).
To start, we need to find three pairs of faces whose areas sum up to 92 square inches:
1. The top and bottom faces of the box have the same dimensions (L x W) and each contribute (L x W) to the surface area.
2. The front and back faces have the same dimensions (L x H) and each contribute (L x H) to the surface area.
3. The left and right faces have the same dimensions (W x H) and each contribute (W x H) to the surface area.
So, we have the following equations:
L x W + L x W + L x H + L x H + W x H + W x H = 184
Simplifying this equation, we get:
2LW + 2LH + 2WH = 184
Now, we know that the sum of the three areas is 92, so we can write the equation:
2LW + 2LH + 2WH = 92
To make the problem more manageable, we can divide everything by 2:
LW + LH + WH = 46
Now, we have an equation that can help us find the dimensions of the box.
To find the possible dimensions, we can use trial and error or solve for one variable in terms of the other two. Let's solve for H:
H = (46 - LW) / (L + W)
Now, we can substitute different values for L and W and calculate the corresponding value for H. By trying different combinations, we can find valid dimensions that satisfy the equation.
For example, let's try L = 4 and W = 6:
H = (46 - 4 x 6) / (4 + 6) = 22 / 10 = 2.2
However, since the dimensions of a box should be integers, this combination does not give a valid solution.
We would need to continue trying different combinations until we find a set of dimensions (L, W, H) that satisfy the equation LW + LH + WH = 46 and are integers.