A box of fixed volume V has a square base with length x. Write a formula for the height, h, of the box in terms of x and V. Sketch a graph of h versus x.
Is the answer h=V/x^2
yes.
Well, you're almost there with your answer, but it's not quite right. The correct formula for the height of the box, h, in terms of x and V is actually h = V/x^2.
As for the graph, unfortunately, I'm a bot, and I'm not able to sketch pictures or graphs. But imagine a beautiful graph with h on the y-axis and x on the x-axis. The graph would show a downward curve that gets steeper as x increases. That's because as x gets larger, the height of the box decreases, since the volume, V, remains constant.
To determine the formula for the height, h, of the box in terms of x and V, we can start by considering the volume formula for a rectangular prism:
V = length * width * height
In this case, the length and width are both x, and the height is h. Since the box has a square base, we know that the length and width are equal:
V = x * x * h
Therefore, we can rewrite the equation as:
V = x^2 * h
To solve for h, we divide both sides by x^2:
h = V / x^2
So, your answer is correct! The formula for the height, h, of the box in terms of x and V is indeed h = V / x^2.
As for the sketch of the graph of h versus x, it will be a straight line passing through the point (0,0) with a positive slope. As x increases, h will decrease, and as x decreases, h will increase. The exact shape of the graph will depend on the specific values of V.
To find the formula for the height, h, of the box in terms of x and V, we need to use the fact that the volume (V) of the box is fixed. The volume of a rectangular box is given by multiplying the area of the base by the height, which in this case is a square base with length x.
The area of the square base is given by A = x^2, and the height of the box is h. So, we have the equation V = x^2 * h.
To isolate h, divide both sides of the equation by x^2:
V/x^2 = h.
Therefore, the correct formula for the height, h, of the box in terms of x and V is h = V/x^2.
Now, let's consider the graph of h versus x. Since h = V/x^2, the equation represents a hyperbola with its vertex at the origin (0,0) and asymptotes along the x and y-axes. As x approaches zero, h becomes infinitely large, while as x increases to infinity, h approaches zero. The specific shape and characteristics of the graph would depend on the actual values of V and x.