An architect needs to design a rectangular room with an area of 77ft^2. What dimensions should he use in order to minimize the perimeter?
I keep getting stuck on this problem. I know when I take square root of 77, it gives me 8.77. But how do I get there.
xy= 77, y= 77/x
P= 2x+2y, P= 2x+2(77/x), P'= 2+154/x
then I have x^2= 154/2
you have
P = 2 x + 154/x
BUT you took the derivative incorrectly
P' = 2 -154/x^2
so
for min or max
0 = 2 - 154/x^2
or
x^2 = 77
NOTE d/dx (1/x) = -1/x^2
To find the dimensions for the rectangular room that will minimize the perimeter, you can follow these steps:
1. Start with the equation for the area of a rectangle, which is equal to the length multiplied by the width: A = length × width.
2. In this case, the area is 77 square feet, so we have the equation: 77 = length × width.
3. Since we want to minimize the perimeter, let's express the perimeter in terms of a single variable, x. We can choose either the length or the width, but let's say x represents the width.
4. Rewrite the equation by solving for length in terms of width: length = 77 / width.
5. The perimeter of a rectangle is given by the formula: P = 2(length + width).
6. Substitute the expressions for length and width into the perimeter formula: P = 2(77 / width + width).
7. Simplify the expression for the perimeter: P = 154 / width + 2x.
8. To find the minimum perimeter, we need to find the value of x for which the derivative of the perimeter with respect to x is equal to zero.
9. Take the derivative of the perimeter equation with respect to x: P' = -154 / width^2 + 2.
10. Set the derivative equal to zero and solve for width: -154 / width^2 + 2 = 0.
11. Multiply both sides of the equation by width^2: -154 + 2width^2 = 0.
12. Add 154 to both sides of the equation: 2width^2 = 154.
13. Divide both sides of the equation by 2: width^2 = 77.
14. Take the square root of both sides to solve for width: width = sqrt(77).
15. Since the length is equal to 77 / width, the length of the rectangular room is: length = 77 / sqrt(77).
So, the architect should use the dimensions width = sqrt(77) and length = 77 / sqrt(77) to minimize the perimeter.
To solve this problem, you are trying to find the dimensions of the rectangular room that will minimize its perimeter.
Let's start by assigning variables to the dimensions of the room. Let's call the length of the room "x" and the width of the room "y". Since the area of the room is given as 77ft^2, we can write the equation:
xy = 77
To minimize the perimeter, we need to find the minimum value of the perimeter function P = 2x + 2y. We can substitute the value of y from the first equation into the perimeter equation to eliminate one variable:
P = 2x + 2(77/x)
To find the minimum value, we need to find the critical points by taking the derivative of the perimeter equation and setting it equal to zero:
P' = 2 - 154/x^2
Setting P' equal to zero, we have:
2 - 154/x^2 = 0
Multiplying through by x^2, we get:
2x^2 = 154
Dividing both sides by 2, we have:
x^2 = 77
Taking the square root of both sides, we get:
x = sqrt(77)
Therefore, the length of the room that will minimize the perimeter is approximately 8.77ft. To find the width of the room, we can substitute this value of x back into the first equation:
8.77y = 77
Solving for y, we get:
y = 77/8.77
y ≈ 8.77ft as well.
So, the architect should design a rectangular room with dimensions of approximately 8.77ft by 8.77ft in order to minimize the perimeter.