Here's the question: Jose needs to enclose a rectangular section of his yard. The area is 35 sq. ft. and the perimeter is 27 sq.ft. Find the length and width of the section.
The issue I'm having is that I think I need to find the common fators of 35 and 27 and they have no common factors so I have no idea what the answer is. Can anyone help?
L*W=35
2L + 2W= 27
From the first, L= 35/W put that in the second.
2*35/W + 2W= 27
multipy through by W, and put in quadratic form, and use the quadratic equation or factor.
To solve the problem, we can follow these steps:
1. We have two equations:
- Equation 1: L * W = 35
- Equation 2: 2L + 2W = 27
2. From Equation 1, we can rearrange it to solve for L in terms of W:
- L = 35 / W
3. Substitute this value of L into Equation 2:
- 2(35 / W) + 2W = 27
4. Simplify the equation:
- 70 / W + 2W = 27
5. Multiply through by W to eliminate the fraction:
- 70 + 2W^2 = 27W
6. Rearrange the equation to quadratic form:
- 2W^2 - 27W + 70 = 0
7. Now, we can either factor the quadratic equation or use the quadratic formula to find the values of W. I'll show you both methods:
Factoring Method:
- We need to find two numbers that multiply to 2 * 70 = 140 and add up to -27. After some trial and error, we find that -20 and -7 satisfy this condition.
- Rewrite the quadratic equation by splitting the middle term:
2W^2 - 20W - 7W + 70 = 0
- Factor by grouping:
2W(W - 10) - 7(W - 10) = 0
(2W - 7)(W - 10) = 0
- Equate each factor to zero and solve for W:
2W - 7 = 0 or W - 10 = 0
W = 7/2 or W = 10
Quadratic Formula Method:
- The quadratic formula is given by:
W = (-b ± √(b^2 - 4ac)) / (2a)
- In our quadratic equation, a = 2, b = -27, and c = 70.
- Substitute these values into the quadratic formula:
W = (-(-27) ± √((-27)^2 - 4 * 2 * 70)) / (2 * 2)
W = (27 ± √(729 - 560)) / 4
W = (27 ± √169) / 4
W = (27 ± 13) / 4
- Solve for both roots:
W = (27 + 13) / 4 or W = (27 - 13) / 4
W = 40 / 4 or W = 14 / 4
W = 10 or W = 7/2
8. Therefore, the possible values for the width (W) are 10 and 7/2.
9. To find the corresponding lengths (L) for each width, substitute these values back into Equation 1:
- For W = 10, L = 35 / 10 = 3.5
- For W = 7/2, L = 35 / (7/2) = 10
10. So, the length and width of the rectangular section are either 3.5 ft by 10 ft or 10 ft by 7/2 ft.