The area of the bottom of a rectangular box is 316cm squared the area of one side is 168cm squared and the area of the other is 120cm squared. What are the dimensions of the box?
If the box's dimensions are x,y,z, then we have
xy = 316
xz = 168
yz = 120
Hmmm. Really? I suspect a typo, since the dimensions are not integers.
316 = 2^2 79
168 = 2^3 3 7
120 = 2^3 3 5
That 79 is a problem.
Let h=height of the box.
Then
width=120/h
length=168/h
We know the area of the base is
Ab=316
so Ab=width*length, or
316=(120/h)*(168/h)=(120*168)/h².
Solve for h
h=√(316/(120*168)
=12√35/√79.
Use the above equations above to find length and width.
To determine the dimensions of the rectangular box, we need to find the length, width, and height.
Let's assume:
Length = L
Width = W
Height = H
Given:
Area of the bottom = 316 cm^2 → L * W = 316 ----(equation 1)
Area of one side = 168 cm^2 → L * H = 168 ----(equation 2)
Area of the other side = 120 cm^2 → W * H = 120 ----(equation 3)
Now, we can solve these equations simultaneously to find the values of L, W, and H.
1. We can solve equation 2 and equation 3 to eliminate H:
L * H = 168 ----(equation 2)
W * H = 120 ----(equation 3)
Multiply equation 2 by W and equation 3 by L to get:
L * W * H = 168W ----(equation 4)
L * W * H = 120L ----(equation 5)
Since we know the product of L * W is equal to 316, we can substitute the values:
316H = 168W ----(equation 6)
316H = 120L ----(equation 7)
2. Divide equation 6 by equation 7:
(316H)/(316H) = (168W)/(120L)
1 = (7W)/(5L)
Cross-multiplying:
5L = 7W
W = (5L) / 7 ----(equation 8)
3. Substitute equation 8 in equation 1 to solve for L:
L * (5L/7) = 316
5L^2 = 2212
L^2 = 442.4
L = sqrt(442.4) = 21.03 cm
4. Substitute the value of L in equation 8 to solve for W:
W = (5L) / 7
W = (5 * 21.03) / 7
W = 15.01 cm
5. Find H by substituting the values of L and W in equation 2:
L * H = 168
21.03 * H = 168
H = 168 / 21.03
H = 7.99 cm
Therefore, the dimensions of the rectangular box are:
Length (L) = 21.03 cm
Width (W) = 15.01 cm
Height (H) = 7.99 cm
To find the dimensions of the rectangular box, we can use the information given about its areas.
Let's assume the length, width, and height of the box are L, W, and H, respectively.
From the given information, we know:
The area of the bottom of the box is L * W = 316 cm² (Equation 1)
The area of one side of the box is L * H = 168 cm² (Equation 2)
The area of the other side of the box is W * H = 120 cm² (Equation 3)
To find the dimensions, we need to solve these three equations simultaneously.
First, let's solve Equation 1 for L:
L * W = 316
L = 316 / W (Equation 4)
Next, we'll substitute Equation 4 into Equation 2:
L * H = 168
(316 / W) * H = 168
By rearranging this equation, we can solve for H:
H = (168 * W) / 316 (Equation 5)
Finally, substitute Equation 4 and Equation 5 into Equation 3 to solve for W:
W * H = 120
W * [(168 * W) / 316] = 120
Simplifying this equation gives us:
W² = (120 * 316) / 168
By taking the square root of both sides, we find the value of W:
W = √[(120 * 316) / 168]
Once we have the value of W, we can substitute it back into Equation 4 to find L:
L = 316 / W
By solving these equations, you can find the values of L, W, and H, which will give you the dimensions of the rectangular box.