A box has rectangular sides, top and bottom. The volume of the box is 3 cubic meters. The height of the box is half the width of the base. Express the total surface area of the box in terms of the height of the box.
Well, this seems like a box-y situation! Let's do some math, shall we?
Let's start by figuring out the dimensions of the box. We know that the volume of the box is 3 cubic meters, and the height of the box is half the width of the base.
So, let's call the width of the base "w" and the height of the box "h". We can express the volume of the box as:
V = w * w * h = 3
Since the height of the box is half the width of the base, we can write:
h = 0.5w
Now, let's express the total surface area of the box in terms of the height "h".
The total surface area of the box consists of the areas of the six rectangular sides. We have two rectangles for the top and bottom, and four rectangles for the sides.
The area of each rectangle can be calculated as length * width. For the top and bottom, the length is equal to the width of the base (w), and the width is equal to the height of the box (h). So, each top and bottom rectangle has an area of w * h.
For the sides, there are two pairs of rectangles. Each pair consists of two rectangles with lengths equal to the width of the base (w) and widths equal to the height of the box (h). So, each pair has an area of 2w * h.
Adding up the areas of all six rectangles, we get:
Total Surface Area = 2(w * h) + 2w * h + 2w * h
Simplifying that, we get:
Total Surface Area = 6w * h
Now, let's substitute the value of h in terms of w:
Total Surface Area = 6w * (0.5w)
That simplifies further to:
Total Surface Area = 3w^2
So, the total surface area of the box in terms of the height of the box (h) is 3 times the square of the width of the base (w).
Hope that puts a smile on your face!
To find the total surface area of the box, we need to calculate the areas of all the sides and add them together.
Let's denote the width of the base as "w" and the height of the box as "h." We know that the volume of the box is 3 cubic meters, so we have:
Volume = width * height * length
Since the box is rectangular, the length is equal to the width. Therefore, we can rewrite the equation as:
3 = w * h * w
We also know that the height of the box is half the width of the base, so we can rewrite the equation as:
3 = w * (w / 2) * w
3 = (w^2 / 2) * w
6 = w^3
w = cbrt(6)
Now that we have the value for w, we can calculate the height (h) using the relationship given in the problem:
h = (1/2) * w
h = (1/2) * cbrt(6)
Next, let's calculate the areas of the sides of the box. There are three pairs of sides: front and back, left and right, top and bottom.
1. Front and Back:
The area of each side is equal to the width (w) multiplied by the height (h):
Front and Back area = 2 * w * h
2. Left and Right:
The area of each side is equal to the length (w) multiplied by the height (h):
Left and Right area = 2 * w * h
3. Top and Bottom:
The area of each side is equal to the length (w) multiplied by the width (w):
Top and Bottom area = 2 * w * w
Now, let's add up the areas of all the sides to find the total surface area:
Total surface area = Front and Back area + Left and Right area + Top and Bottom area
Total surface area = 2 * w * h + 2 * w * h + 2 * w * w
Total surface area = 4 * w * h + 2 * w^2
Substituting the values we calculated for w and h:
Total surface area = 4 * cbrt(6) * (1/2 * cbrt(6)) + 2 * (cbrt(6))^2
Total surface area = 2 * (cbrt(6))^2 + 4 * (cbrt(6))^3
Therefore, the total surface area of the box in terms of the height of the box (h) is 2 * (cbrt(6))^2 + 4 * (cbrt(6))^3.
To express the total surface area of the box in terms of its height, we need to find the dimensions of the box.
Let's start by assuming the width of the base is "w." According to the given information, the height of the box is half the width, so the height can be represented as "h = (1/2)w".
The volume of the box is given as 3 cubic meters, which can be expressed as "V = lwh". Since the box has rectangular sides, top, and bottom, the length of the box can also be represented as "l = w". Therefore, we can rewrite the volume equation as "3 = w * w * h".
Now, substitute the value of h into the rewritten volume equation:
3 = w * w * (1/2)w
3 = (1/2)w^3
To find the width of the box, we can solve for w:
w^3 = 2 * 3
w^3 = 6
w ≈ 1.817
Now that we have the width of the base, we can find the height by substituting the value of w into the equation for h:
h = (1/2) * 1.817
h ≈ 0.909
To calculate the total surface area of the box, we need to consider all six sides: the top, bottom, front, back, left, and right.
Top and bottom: Each of these surfaces is a rectangle with dimensions w x w, so the total area for both the top and bottom is given by 2 * w^2.
Front and back: These surfaces are also rectangles with dimensions w x h, so the total area for both the front and back is given by 2 * w * h.
Left and right: Similarly, these surfaces are rectangles with dimensions h x w, so the total area for both the left and right sides is given by 2 * h * w.
Adding up all six sides, we get the total surface area:
Total Surface Area = 2 * w^2 + 2 * w * h + 2 * h * w
Substituting the values we found earlier for w and h:
Total Surface Area ≈ 2 * (1.817)^2 + 2 * 1.817 * 0.909 + 2 * 0.909 * 1.817
Simplifying this expression would provide an answer for the total surface area of the box in terms of the height.
If the height is h, then the base has dimensions 2h and y, so
2h^2y = 3
so y = 3/(2h^2)
and the area is two pairs of vertical faces:
2(2h*h + y*h) = 2(2h^2+3/(2h^2)*h)
= 4h^2 + 3/h