The areas of the faces of a rectangular box are 48 m2, 96 m2, and 288 m2.
A second box is cubical and each of its faces has area 16 m2.
Find the ratio of the volume of the first box to the volume of the second box.
A) 18 : 1 B) 12 : 1 C) 324 : 1 D) 16 : 1
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The areas of the faces of a rectangular box are 8 m2, 12 m2 and 6 m2. Find its volume.
To find the ratio of the volume of the first box to the volume of the second box, we need to find the volumes of both boxes.
For the first box, we can find the dimensions of the rectangular box by finding the lengths of the sides. Let's denote the lengths as x, y, and z, where x, y, and z are the lengths of the sides of the box.
From the given areas, we can write the following equations:
xy = 48 (Equation 1)
xz = 96 (Equation 2)
yz = 288 (Equation 3)
To find the lengths of the sides, we need to solve these equations simultaneously. We can start by dividing Equation 3 by Equation 1:
(yz) / (xy) = 288 / 48
y / x = 6 (Equation 4)
Similarly, dividing Equation 3 by Equation 2 gives:
(z / x) = 288 / 96
z / x = 3 (Equation 5)
Combining Equations 4 and 5, we get:
y / x * z / x = 6 * 3
(yz) / (x^2) = 18
yz = 18x^2
Substituting this into Equation 2, we have:
18x^2 = 96
x^2 = 96 / 18
x^2 = 16
x = 4
Substituting the value of x back into Equation 1, we get:
4y = 48
y = 48 / 4
y = 12
And substituting it back into Equation 5, we get:
z = 3x
z = 3 * 4
z = 12
So, the dimensions of the first box are x = 4, y = 12, and z = 12.
Now, let's find the volume of the first box by multiplying its dimensions:
Volume of the first box = x * y * z = 4 * 12 * 12 = 576 m^3
For the second box, it is stated that each face has an area of 16 m^2. Since it is a cube, all sides have the same length, denoted as l. So, the area of each face is given by l^2 = 16. Solving for l, we have:
l^2 = 16
l = √16
l = 4
Therefore, the length of each side of the second box is 4 m.
The volume of the second box is given by:
Volume of the second box = l^3 = 4^3 = 64 m^3
To find the ratio of the volume of the first box to the volume of the second box, we divide the volume of the first box by the volume of the second box:
Ratio = (Volume of the first box) / (Volume of the second box) = 576 / 64
Simplifying the ratio, we get:
Ratio = 9
So, the ratio of the volume of the first box to the volume of the second box is 9:1.
Therefore, the answer is not listed among the provided options.
To solve this problem, we need to find the volume of both boxes and compare them to find the ratio between the two volumes.
Let's start with the first box, which is a rectangular box. We are given the areas of its faces, which are 48 m², 96 m², and 288 m². The formula to find the area of a rectangle is length multiplied by width.
Let's assume the length, width, and height of the first box are l₁, w₁, and h₁, respectively. We know that:
l₁ * w₁ = 48 (Equation 1)
l₁ * h₁ = 96 (Equation 2)
w₁ * h₁ = 288 (Equation 3)
To solve for the dimensions of the first box, we can rearrange these equations:
l₁ = 48 / w₁
h₁ = 96 / l₁
h₁ = 96 / (48 / w₁)
h₁ = 2w₁
Substituting this value of h₁ back into Equation 3, we have:
w₁ * (2w₁) = 288
2w₁² = 288
w₁² = 144
w₁ = √(144)
w₁ = 12
Now substituting the value of w₁ back into Equation 1, we can solve for l₁:
l₁ * 12 = 48
l₁ = 48 / 12
l₁ = 4
And finally, substituting the values of l₁ and w₁ into Equation 2, we can find h₁:
4 * h₁ = 96
h₁ = 96 / 4
h₁ = 24
So, the dimensions of the first box are l₁ = 4, w₁ = 12, and h₁ = 24.
Now let's move on to the second box, which is a cube. All its faces have the same area, which is 16 m². The formula to find the area of a square (which is what each face of the cube is) is side length squared.
Let the side length of the second box be s₁. We know that s₁² = 16, so s₁ = √(16) = 4.
Since the second box is a cube, all sides have the same length, which is 4.
To find the volume of each box, we can use the formula for rectangular and cubic volume:
For the first box: Volume₁ = length * width * height = l₁ * w₁ * h₁
For the second box: Volume₂ = side length * side length * side length = s₁ * s₁ * s₁
Plugging in the values we found:
Volume₁ = 4 * 12 * 24 = 1152
Volume₂ = 4 * 4 * 4 = 64
Therefore, the ratio of the volume of the first box to the volume of the second box is 1152 : 64, which simplifies to 18 : 1.
So, the answer is A) 18 : 1.