A rectangular prism has length x+2 width x+1, height of 4, and a volume of 24. Find the length and width.
12
To solve this problem, we can use the formula for the volume of a rectangular prism, which is:
Volume = Length × Width × Height
Given that the volume is 24, the height is 4, and the length is x+2, we can substitute these values into the formula:
24 = (x+2) × Width × 4
Now, let's simplify the equation:
24 = 4x + 8 × Width
Next, we need to find the value of Width. To do that, we can divide both sides of the equation by 8:
3 = x + 2 × Width
Now, we can rearrange the equation to solve for Width:
Width = (3 - x) / 2
Therefore, the width of the rectangular prism is (3 - x) / 2.
To find the length, we can substitute the width value we just found into the volume formula:
24 = (x+2) × [(3 - x) / 2] × 4
Simplifying the equation further:
24 = 2(x+2)(3 - x)
Now, let's expand the equation:
24 = 2(3x - x^2 + 6 - 2x)
24 = 2(9 - x^2 + 4x)
24 = 18 - 2x^2 + 8x
Combining like terms:
0 = -2x^2 + 8x - 6
Let's divide the entire equation by -2 to simplify further:
0 = x^2 - 4x + 3
Now, we can factor the quadratic equation:
0 = (x-3)(x-1)
Setting each factor equal to zero, we get:
x - 3 = 0 or x - 1 = 0
Solving for x, we find:
x = 3 or x = 1
Therefore, the possible values for the length (x+2) are:
Length = 3 + 2 = 5
Length = 1 + 2 = 3
Similarly, substituting these values of x in the width equation:
Width = (3 - x) / 2
When x = 3, width = (3 - 3) / 2 = 0 / 2 = 0
When x = 1, width = (3 - 1) / 2 = 2 / 2 = 1
So, the possible values for the width are:
Width = 0
Width = 1
Therefore, the length and width of the rectangular prism can be 5 and 0 respectively, or 3 and 1 respectively.
To find the length and width of the rectangular prism, we can use the given information and the volume formula of a rectangular prism.
The volume of a rectangular prism is given by: Volume = Length * Width * Height
Given that the volume is 24 and the height is 4, we can substitute these values into the formula and solve for the length and width.
24 = (x + 2) * (x + 1) * 4
Let's start by dividing both sides of the equation by 4:
6 = (x + 2) * (x + 1)
Now, let's expand the equation:
6 = x^2 + 3x + 2
Rearrange the equation to make it a quadratic equation:
x^2 + 3x + 2 - 6 = 0
x^2 + 3x - 4 = 0
Now, we can either solve this quadratic equation by factoring, completing the square, or by using the quadratic formula. Let's use factoring.
The equation can be factored into:
(x + 4)(x - 1) = 0
Setting each factor equal to zero gives us:
x + 4 = 0 or x - 1 = 0
Solving each equation gives us:
x = -4 or x = 1
Since the length and width of a rectangular prism cannot be negative, we can disregard x = -4.
Hence, the length of the rectangular prism is x + 2 = 1 + 2 = 3, and the width is x + 1 = 1 + 1 = 2.
Therefore, the length is 3 and the width is 2.
4(x+1)(x+2) = 24
even without solving the equation, you know that
(x+1)(x+2) = 6
so, what two consecutive numbers multiply to 6?