A rectangular prism has a width of 92 ft and a volume of 240 ft3 find the volume of a similar prism with a width of 46 ft round to the nearest tenth if necessary
1. 30 ft3
2. 40 ft3
3. 60 ft3
4. 120 ft3
4. 120 ft3
Explanation:
Since the two prisms are similar, their corresponding dimensions are proportional.
Let the length, height, and length of the second prism be x, y, and z respectively.
Then, we have:
x : 46 = y : y/2 = z : 92
Simplifying each ratio, we get:
x : 46 = y : 2y = z : 92
From the given information, we know that the volume of the first prism is 240 ft3.
Volume of first prism = width x height x length
240 = 92 x y x z
Solving for y and z, we get:
y = 240/(92 x z)
z = 240/(92 x y)
Substituting these values into the ratio equation, we get:
x : 46 = (240/(92 x z)) : (480/(92 x z)) = (240/(92 x y)) : (92/(92 x y))
Simplifying, we get:
x : 46 = 5 : 10
Therefore, x = 23.
Now we can find the volume of the second prism:
Volume of second prism = width x height x length
= 46 x (2y) x 23
= 92y x 23
Substituting y = 240/(92 x z) and z = 240/(92 x y) into the formula, we get:
Volume of second prism = 240 x 23
= 5520
Rounding to the nearest tenth, we get 5520.0, which is approximately 120 ft3.
To find the volume of a similar prism, we need to use the concept of scale factor. The scale factor is the ratio of the corresponding sides of two similar objects.
Given that the width of the first prism is 92 ft and its volume is 240 ft^3, we can set up the following ratio between the widths and volumes of the two prisms:
Width ratio = 46 ft / 92 ft = 1/2
Volume ratio = V2 / 240 ft^3
Since the width ratio is 1/2, the volume ratio will be (1/2)^3 = 1/8. This is because volume is proportional to length, width, and height, and the ratio between corresponding sides is cubed.
Now, we can set up the equation to find the volume of the second prism:
V2 / 240 ft^3 = 1/8
To solve for V2, we can cross multiply:
V2 = (1/8) * 240 ft^3
V2 = 30 ft^3
Therefore, the volume of the similar prism with a width of 46 ft is approximately 30 ft^3.
So, the correct answer is 1. 30 ft^3.