The height of a square-based box is 4 cm more than the side length of its square base. If the volume of the box is 225 cm3. What are its dimensions?
So this is what I did:
v = l x w x h
v = 1x x 1x x (x+4)
v = 2x^2 + 8x
225 = 2x^2 + 8x
2x^2 + 8x - 225 = 0
Then I did the quadratic formula and got 8.8 as an x.
So I assume the side lengths are 8.8 and the height would be 8.8+4 so it would be 12.8.
I'm not sure if I did the equation correctly though. Thank you!
First, Online, “*” is used to indicate multiplication to avoid confusion with “x” as an unknown.
V = x * x * (x+4)
You are adding instead of multiplying.
V = x^2 * (x+4) = x^3 + 4x^2
Well, it seems like you did a pretty good job with the equation! However, I must say that the value of x you got, 8.8, seems a bit off. Let me crunch the numbers and see if I can help you out.
Using the equation you set up, 2x^2 + 8x - 225 = 0, we can solve for x. Plugging it into the quadratic formula, we get:
x = (-8 ± √(8^2 - 4(2)(-225))) / (2(2))
Simplifying this gives us:
x = (-8 ± √(64 + 1800)) / 4
x = (-8 ± √1864) / 4
Now, we need to find the positive value of x, since negative lengths don't make sense in this context. So let's evaluate that:
x ≈ ( -8 + √1864) / 4
x ≈ 10.8 / 4
x ≈ 2.7
So, the side length of the square base is approximately 2.7 cm. Now, if the height is 4 cm more than the side length, it would be:
Height ≈ 2.7 + 4
Height ≈ 6.7 cm
Therefore, the dimensions of the square-based box are approximately 2.7 cm x 2.7 cm x 6.7 cm.
To solve for the dimensions of the box, let's first correct the equation you wrote:
v = l x w x h
In this case, the box has a square base, so the length (l) and width (w) will be the same. Therefore, we can replace both l and w with the side length of the square base (x).
Now, we can rewrite the equation:
225 = x * x * (x + 4)
Simplifying further:
225 = x^2 * (x + 4)
To solve for x, we can rearrange the equation:
x^3 + 4x^2 - 225 = 0
Using the quadratic formula, with a = 1, b = 4, and c = -225:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-4 ± √(4^2 - 4(1)(-225))) / (2(1))
Simplifying:
x = (-4 ± √(16 + 900)) / 2
x = (-4 ± √916) / 2
x ≈ (-4 ± 30.27) / 2
Now, we need to determine the positive solution since we're working with dimensions. Neglecting the negative solution, we have:
x ≈ (-4 + 30.27) / 2
x ≈ 26.27 / 2
x ≈ 13.14
So, the side length of the square base is approximately 13.14 cm.
Finally, to find the height, we need to add 4 cm to the side length:
height ≈ 13.14 + 4
height ≈ 17.14 cm
Therefore, the dimensions of the box are approximately 13.14 cm (side length) by 13.14 cm (side length) by 17.14 cm (height).
To find the dimensions of the square-based box, we can use the given information and set up an equation based on the volume formula.
Let's denote the side length of the square base as "x" and the height as "x + 4" since the height is 4 cm more than the side length.
Using the volume formula for a rectangular prism, we have:
Volume = length × width × height
Since the base of the box is a square, the length and width of the base are both "x". So the equation becomes:
225 cm³ = x × x × (x + 4)
Simplifying further:
225 = x² × (x + 4)
Now, let's solve this equation for x.
Expanding the equation:
225 = x³ + 4x²
Rearranging the equation:
x³ + 4x² - 225 = 0
Now, to solve this cubic equation, you can either:
1. Use a graphing calculator or software to find the root of the equation.
2. Use numerical methods such as the Newton-Raphson method to approximate the roots.
3. Factor the equation (if possible).
In this case, factoring the equation might not be straightforward. So, let's use a calculator or software to find the roots of the cubic equation, or approximate the root using numerical methods.
Assuming you have found the approximate root to be x = 8.8, you can substitute this value back into the equation to find the height:
h = x + 4
h = 8.8 + 4
h = 12.8 cm
Therefore, the dimensions of the square-based box would be:
- Side length of the square base: 8.8 cm
- Height of the box: 12.8 cm