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Answer :
Sure! Let's go through the solution step-by-step:
1. Understand the problem:
- We have a rectangular parking lot.
- The length of the parking lot is 7 yards greater than its width.
- The area of the parking lot is 120 square yards.
- We need to find both the length and width of the parking lot.
2. Set up the equation:
- Let's call the width of the parking lot [tex]\( w \)[/tex] yards.
- According to the problem, the length would then be [tex]\( w + 7 \)[/tex] yards.
- The area of a rectangle is found by multiplying the length by the width. Therefore, the equation for the area is:
[tex]\[
w \times (w + 7) = 120
\][/tex]
3. Solve the equation:
- Expand the area equation:
[tex]\[
w^2 + 7w = 120
\][/tex]
- Rearrange it into a standard quadratic equation format:
[tex]\[
w^2 + 7w - 120 = 0
\][/tex]
- Solve this quadratic equation for [tex]\( w \)[/tex].
4. Apply the quadratic formula:
- The quadratic formula is given by:
[tex]\[
w = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
- For our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -120 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 7^2 - 4 \times 1 \times (-120) = 49 + 480 = 529
\][/tex]
- Take the square root of the discriminant:
[tex]\[
\sqrt{529} = 23
\][/tex]
- Calculate the two possible solutions:
[tex]\[
w = \frac{{-7 \pm 23}}{2}
\][/tex]
[tex]\[
w = \frac{{-7 + 23}}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
w = \frac{{-7 - 23}}{2} = \frac{-30}{2} = -15
\][/tex]
- Since width cannot be negative, we take the positive solution, [tex]\( w = 8 \)[/tex].
5. Find the length:
- Since the length is [tex]\( 7 \)[/tex] yards more than the width, the length is:
[tex]\[
8 + 7 = 15
\][/tex]
6. Conclusion:
- The parking lot has a width of 8 yards.
- The parking lot has a length of 15 yards.
These dimensions satisfy the conditions given in the problem, as the area calculates to 120 square yards, which matches perfectly.
1. Understand the problem:
- We have a rectangular parking lot.
- The length of the parking lot is 7 yards greater than its width.
- The area of the parking lot is 120 square yards.
- We need to find both the length and width of the parking lot.
2. Set up the equation:
- Let's call the width of the parking lot [tex]\( w \)[/tex] yards.
- According to the problem, the length would then be [tex]\( w + 7 \)[/tex] yards.
- The area of a rectangle is found by multiplying the length by the width. Therefore, the equation for the area is:
[tex]\[
w \times (w + 7) = 120
\][/tex]
3. Solve the equation:
- Expand the area equation:
[tex]\[
w^2 + 7w = 120
\][/tex]
- Rearrange it into a standard quadratic equation format:
[tex]\[
w^2 + 7w - 120 = 0
\][/tex]
- Solve this quadratic equation for [tex]\( w \)[/tex].
4. Apply the quadratic formula:
- The quadratic formula is given by:
[tex]\[
w = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
- For our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -120 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 7^2 - 4 \times 1 \times (-120) = 49 + 480 = 529
\][/tex]
- Take the square root of the discriminant:
[tex]\[
\sqrt{529} = 23
\][/tex]
- Calculate the two possible solutions:
[tex]\[
w = \frac{{-7 \pm 23}}{2}
\][/tex]
[tex]\[
w = \frac{{-7 + 23}}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
w = \frac{{-7 - 23}}{2} = \frac{-30}{2} = -15
\][/tex]
- Since width cannot be negative, we take the positive solution, [tex]\( w = 8 \)[/tex].
5. Find the length:
- Since the length is [tex]\( 7 \)[/tex] yards more than the width, the length is:
[tex]\[
8 + 7 = 15
\][/tex]
6. Conclusion:
- The parking lot has a width of 8 yards.
- The parking lot has a length of 15 yards.
These dimensions satisfy the conditions given in the problem, as the area calculates to 120 square yards, which matches perfectly.
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