A rectangular prism has a square base with area 9 m2. The surface area of the prism is 78 m2. What are the dimensions of the prism?
the base has side length 3
there are 4 sides with area 3h = 12h
there are two ends with area 9 = 18
12h+18 = 78
h=5
The prism is 3x3x5
Well, well, well, let's crack some jokes while solving this math problem, shall we?
Since the base of this rectangular prism is a square, we know that the length and the width are the same. So, let's call that 'x' meters.
Now, the total surface area of this hilarious prism is 78 square meters.
The surface area of the base (which is a square with side 'x') would be x times x, which is just x squared. But, hey, we already know that it's 9 m2. So, we can solve for x by setting x squared to 9.
x squared = 9
Now, since I can't do math, let's do some magic!
Abracadabra... x = 3 m! Ta-da!
So, the dimensions of this rectangular prism are 3 meters by 3 meters by something we don't know. Oh, what a mystery!
To find the dimensions of the rectangular prism, we first need to determine the length of the sides of the square base.
Given that the area of the square base is 9 m², we can find the side length using the formula for the area of a square:
Area = side length × side length
Rearranging the formula, we have:
side length = √Area
Substituting the given area of 9 m² into the formula, we have:
side length = √9 = 3 m
Since the base of the rectangular prism is square, all sides of the base have a length of 3 m.
Next, we need to determine the dimensions of the rectangular prism. The surface area of a rectangular prism is found by summing the areas of all six faces:
Surface area = 2(length × width) + 2(length × height) + 2(width × height)
We are given that the surface area is 78 m², so we can set up the equation:
78 = 2(length × 3) + 2(length × height) + 2(3 × height)
Simplifying the equation gives:
78 = 6(length) + 2(length × height) + 6(height)
Now we need to find a system of equations to solve for the dimensions.
Since the base is square, the length and width are both 3 m.
From the equation above, we can set up a system of equations by equating the two expressions for the surface area:
6(length) + 2(length × height) + 6(height) = 78
Substituting 3 for the length and simplifying gives:
6(3) + 2(3 × height) + 6(height) = 78
18 + 6(height) + 2(3 × height) = 78
18 + 6(height) + 6(height) = 78
18 + 12(height) = 78
30(height) = 78 - 18
30(height) = 60
height = 60 / 30
height = 2 m
Therefore, the dimensions of the rectangular prism are:
Length = 3 m
Width = 3 m
Height = 2 m
To find the dimensions of the prism, we need to use the given information about the base area and surface area.
Let's start by finding the dimensions of the square base.
Since the base of the prism is a square, let's call the side length of the square base "x".
The area of a square is given by the formula A = side length x side length. We know the area of the square base is 9 m², so we can write the equation as:
9 = x * x
Simplifying this equation, we have:
x² = 9
To solve for x, we can take the square root of both sides:
√(x²) = √(9)
x = 3
So, the side length of the square base is 3 m.
Now, let's find the dimensions of the prism.
The surface area of a rectangular prism consists of six faces. The formula for the surface area of a rectangular prism is:
Surface area = 2(length x width + length x height + width x height)
We need to find three unknowns: length, width, and height.
We know that the surface area of the prism is 78 m². We can substitute this value into the surface area formula and use the known information about the base area to solve for the dimensions.
78 = 2(length x width + length x height + width x height)
Since the base is square with a side length of 3 m, the length and width of the prism are also 3 m.
78 = 2(3 x 3 + 3 x height + 3 x height)
78 = 2(9 + 6h)
Divide both sides by 2 to isolate the expression inside the parenthesis:
39 = 9 + 6h
Subtract 9 from both sides:
30 = 6h
Divide both sides by 6:
5 = h
So, the height of the prism is 5 m.
In summary, the dimensions of the rectangular prism are:
- Length = 3 m
- Width = 3 m
- Height = 5 m