The base of a rectangular prism has dimensions (x)cm by (2x) cm and a height of (x + 2) cm. If the total surface area is 63cm squared, find the value of x and the dimensions of the rectangular prism....
add up the pairs of faces
2(x(2x) + x(x+2) + 2x(x+2)) = 63
solve for x.
You sure you don't have a typo? Since there are two of each face, I'd expect the area to be an even number, like 64, rather than 63.
If the question had the dimensions are whole numbers.
How about x = appr 1.98 ?
http://www.wolframalpha.com/input/?i=2(x(2x)+%2B+x(x%2B2)+%2B+2x(x%2B2))+%3D+63
To find the value of x and the dimensions of the rectangular prism, we need to set up and solve an equation based on the given information.
Let's start by calculating the total surface area of a rectangular prism. The total surface area is the sum of the areas of all six faces. For a rectangular prism, this can be calculated using the formula:
Surface Area = 2lw + 2lh + 2wh
Where l is the length, w is the width, and h is the height of the prism.
In this case, the length (l) is x cm, the width (w) is 2x cm, and the height (h) is (x + 2) cm. We are given that the total surface area is 63 cm^2.
Substituting the given values:
63 = 2(x)(2x) + 2(x)(x + 2) + 2(2x)(x + 2)
Now, let's simplify and solve for x:
63 = 4x^2 + 2x(x + 2) + 4x(x + 2)
63 = 4x^2 + 2x^2 + 4x + 4x^2 + 8x
63 = 10x^2 + 12x
Rearranging the equation:
10x^2 + 12x - 63 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's solve it by factoring:
(2x - 3)(5x + 7) = 0
Setting each factor equal to zero:
2x - 3 = 0 or 5x + 7 = 0
Solving each equation separately:
2x = 3 or 5x = -7
x = 3/2 or x = -7/5
Since lengths cannot be negative, we can discard the x = -7/5 solution. Therefore, x = 3/2.
Now that we have the value of x, we can find the dimensions of the rectangular prism.
Length (l) = x = 3/2 cm
Width (w) = 2x = 2(3/2) = 3 cm
Height (h) = x + 2 = 3/2 + 2 = 7/2 cm
So, the dimensions of the rectangular prism are 3/2 cm, 3 cm, and 7/2 cm.