elaine and Daniel are building a rectangular greenhouse.they want the are of the floor to be 36metres squared.since the glass walls are expensive,they want to minimise the amount of glass wall they use.they have commissioned you to design a greenhouse which minimizes the cost of the glass walls.if the greenhouse is to have a height of 2metre and the price of the glass from clear view glass is R46.75,what will it cost to purchase the glass for the walls of the greenhouse?
To minimize the cost of the glass walls, we need to find the dimensions of the rectangular greenhouse that have the smallest perimeter (which corresponds to the smallest amount of glass needed).
Let's assume the length of the greenhouse is L meters and the width is W meters.
Given that the area of the floor is 36 square meters and the height of the greenhouse is 2 meters, we can write the equation: L * W = 36.
To minimize the perimeter, we can use the formula for the perimeter of a rectangle: 2 * (L + W).
Solving for L in terms of W from the area equation, we have: L = 36 / W.
Substituting this value of L in the equation for the perimeter: P = 2 * (36 / W + W).
To find the minimum value of the perimeter, we can take the derivative with respect to W and set it equal to zero: dP/dW = 0.
Let's find the critical points by differentiating with respect to W:
dP/dW = 2 * (-36 / W^2 + 1).
Setting dP/dW = 0, we have:
-36 / W^2 + 1 = 0.
Solving for W, we find:
36 / W^2 = 1.
W^2 = 36.
Taking the square root of both sides, we have:
W = 6.
Now that we know the value of W, we can substitute it back into the area equation to find L:
L * 6 = 36.
L = 6.
Therefore, the dimensions of the greenhouse that minimize the amount of glass walls required are 6 meters by 6 meters.
Now we can calculate the cost of the glass walls. The perimeter of the greenhouse is:
P = 2 * (6 + 6) = 24 meters.
Since the height of the greenhouse is given as 2 meters, the total area of the glass walls is:
A = P * H = 24 * 2 = 48 square meters.
Finally, the cost to purchase the glass for the walls of the greenhouse can be calculated by multiplying the area of the glass walls by the price of the glass:
Cost = A * Price = 48 * R46.75 = R2244.
Therefore, it will cost R2244 to purchase the glass for the walls of the greenhouse.
To find the cost of the glass walls for the greenhouse, we first need to calculate the area of the four walls.
Since we have the area of the floor and the height, we can use the formula:
Area of the walls = 2 * (length * height + width * height)
We can then rearrange the formula to solve for length:
length = Area of the floor / (2 * height)
Using the given values, we have:
length = 36 / (2 * 2) = 9
Now that we have the length and the height, we can calculate the area of the four walls:
Area of the walls = 2 * (length * height + width * height)
Since the greenhouse is rectangular, the width would also be 9 meters.
Area of the walls = 2 * (9 * 2 + 9 * 2) = 2 * (18 + 18) = 2 * 36 = 72 square meters
Next, we need to find the cost of the glass walls. The price of the glass from Clear View Glass is given as R46.75 per square meter.
So, to calculate the cost of the glass for the walls, we multiply the area of the walls by the price per square meter:
Cost of the glass walls = Area of the walls * Price per square meter
Cost of the glass walls = 72 * R46.75 = R3369
Therefore, it will cost R3369 to purchase the glass for the walls of the greenhouse.