To solve for the length and width of the playground given that the area is 160 square yards and the width is 6 yards longer than the length, let's go through a step-by-step solution:
1. Define Variables:
Let the length of the playground be [tex]\( L \)[/tex] (in yards).
Therefore, the width [tex]\( W \)[/tex] will be [tex]\( L + 6 \)[/tex] (in yards).
2. Formulate the Equation:
Given the area of the playground is 160 square yards, we have the equation:
[tex]\[
\text{Length} \times \text{Width} = \text{Area}
\][/tex]
Substituting the expressions for length and width:
[tex]\[
L \times (L + 6) = 160
\][/tex]
3. Solve the Equation:
Expand the equation:
[tex]\[
L^2 + 6L = 160
\][/tex]
Rearrange it into standard quadratic form:
[tex]\[
L^2 + 6L - 160 = 0
\][/tex]
4. Find Solutions for [tex]\( L \)[/tex]:
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. We can solve this using standard methods such as factoring, the quadratic formula, or completing the square. The solutions to the quadratic equation [tex]\( L^2 + 6L - 160 = 0 \)[/tex] are:
[tex]\[
L_1 = 10 \quad \text{(real and positive solution)}
\][/tex]
5. Determine Width Using Length:
Once we have the length [tex]\( L \)[/tex]:
[tex]\[
L = 10 \text{ yards}
\][/tex]
We can find the width [tex]\( W \)[/tex]:
[tex]\[
W = L + 6 = 10 + 6 = 16 \text{ yards}
\][/tex]
So, the length of the playground is 10 yards, and the width is 16 yards.
Finally, we can verify by plugging these values back into the area formula to ensure they satisfy the given area:
[tex]\[
\text{Length} \times \text{Width} = 10 \times 16 = 160 \text{ square yards}
\][/tex]
This confirms our solution is correct.
Answer: The length of the playground is [tex]\( 10 \text{ yards} \)[/tex] and the width is [tex]\( 16 \text{ yards} \)[/tex].
Thus, the correct option is:
length [tex]\(= 10 \text{ yd}\)[/tex], width [tex]\(= 16 \text{ yd}\)[/tex].
