Question
It is desired to cover a soccer field with grass, whose surface area is (p^{4}-16) square meters, with p a real number greater than 2. If the rolls of grass measure (p-2) meters wide and (p+2) meters long. How many rolls of grass are needed to cover the entire field?
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Answer to a math question It is desired to cover a soccer field with grass, whose surface area is (p^{4}-16) square meters, with p a real number greater than 2. If the rolls of grass measure (p-2) meters wide and (p+2) meters long. How many rolls of grass are needed to cover the entire field?
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1. Determine the area of one roll of grass:
\text{Area of one roll} = (p-2) \times (p+2) = p^2 - 4
2. Determine the surface area of the soccer field:
\text{Field area} = p^4 - 16
3. Determine the number of rolls needed:
\text{Number of rolls} = \frac{\text{Field area}}{\text{Area of one roll}} = \frac{p^4 - 16}{p^2 - 4}
4. Simplify the expression:
\frac{p^4 - 16}{p^2 - 4} = \frac{(p^2 - 4)(p^2 + 4)}{p^2 - 4} = p^2 + 4
5. Therefore, the number of rolls needed:
p^2 + 4
2. Determine the surface area of the soccer field:
3. Determine the number of rolls needed:
4. Simplify the expression:
5. Therefore, the number of rolls needed:
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