An aquarium of height 1.5 feet is to have a volume of 12 ft3. Let x denote the length of the base and y the width.
a) express y as a function of x
b) Express the total number S of square feet of glass needed as a function of x.
volume = area of base * height
so
x y = 12/1.5 = 8 ft^2
y = 8/x
base area = x y = 8
end area = 2 xh = 3 x
side area = 2 y h = 3 y = 3*8/x = 24/x
total area = 8 + 3 x + 24/x
a) Well, let's break this down. The formula for the volume of a rectangular prism is V = lwh, where V is the volume, l is the length, w is the width, and h is the height. In this case, we know that the volume (V) is 12 ft^3 and the height (h) is 1.5 feet.
So, we can write the equation as 12 = x * y * 1.5. To express y as a function of x, we can rearrange this equation by dividing both sides by (1.5 * x):
12 / (1.5 * x) = y.
Simplifying the equation, we get:
8 / x = y.
Therefore, y is a function of x, given by y = 8 / x.
b) Now, let's tackle the second part. To find the total number of square feet of glass needed, we need to calculate the surface area of the aquarium.
The surface area of a rectangular prism can be found using the formula S = 2lw + 2lh + 2wh, where S is the surface area, l is the length, w is the width, and h is the height.
Plugging in the values for l (x), w (y), and h (1.5), we have:
S = 2xy + 2x(1.5) + 2y(1.5).
Simplifying this equation a bit, we get:
S = 2xy + 3x + 3y.
So, the total number S of square feet of glass needed is a function of x, given by S = 2xy + 3x + 3y.
Hope that helps, and remember, stay glassy!
To express y as a function of x, we can use the formula for the volume of a rectangular prism:
Volume = Length x Width x Height
Given that the height is 1.5 feet and the volume is 12 ft3, we have:
12 = x * y * 1.5
Dividing both sides of the equation by 1.5:
12/1.5 = x * y
Simplifying:
8 = x * y
Therefore, the function for y in terms of x is:
y = 8/x
To express the total number S of square feet of glass needed as a function of x, we need to calculate the surface area of the aquarium.
The aquarium has 3 pairs of equal-sized rectangular faces:
- The top and bottom faces have dimensions x * y.
- The two side faces have dimensions 1.5 * x.
- The front and back faces have dimensions 1.5 * y.
The total surface area, S, is given by:
S = 2 * (x * y) + 2 * (1.5 * x) + 2 * (1.5 * y)
Simplifying the expression:
S = 2xy + 3x + 3y
Therefore, the function for S in terms of x is:
S = 2xy + 3x + 3y
To find the expressions for y and the total number S of square feet of glass needed, we need to use the information given about the height, volume, and dimensions of the aquarium.
a) Expressing y as a function of x:
We know that the volume of the aquarium is given by the equation:
Volume = Length × Width × Height
In this case, the volume is given as 12 ft³, and the height is given as 1.5 feet. So we can write:
12 = x × y × 1.5
To express y as a function of x, we can solve this equation for y:
12 = 1.5xy (divided both sides by 1.5)
y = 8/x
Therefore, y can be expressed as a function of x: y = 8/x
b) Expressing the total number S of square feet of glass needed as a function of x:
The aquarium has six sides: two ends, two long sides, and the top and bottom (assuming there is no glass on the back).
The lengths of the two ends are y, and the lengths of the two long sides are x. The top and bottom surfaces both have an area of x * y. So the total number S of square feet of glass needed can be calculated as:
S = 2(y * 1.5) + 2(x * 1.5) + (x * y) + (x * y)
Simplifying this expression, we get:
S = 3y + 3x + 2xy
Substituting the expression for y from part a) into this equation, we have:
S = 3(8/x) + 3x + 2x(8/x)
Simplifying further, we get:
S = 24/x + 3x + 16/x
Therefore, the total number S of square feet of glass needed can be expressed as a function of x: S = 24/x + 3x + 16/x