Gavin is subdividing land into two plots, where one is in the shape of a square and the other is in the shape of a rectangle. The square plot of land has a side length of 4x^3 feet The rectangular plot of land has a length of 2x^5 feet and a width of 6x^2 feet. 1) Use the properties of exponents to determine the simplified expression that represents the area of each plot of land. (2 points for each expression) 2) Then determine which plot of land has the larger area if x=3, how do you know? (2 points for answer, 2 points for explanation)

Question

Gavin is subdividing land into two plots, where one is in the shape of a square and the other is in the shape of a rectangle. The square plot of land has a side length of 4x^3 feet The rectangular plot of land has a length of 2x^5 feet and a width of 6x^2 feet. 1) Use the properties of exponents to determine the simplified expression that represents the area of each plot of land. (2 points for each expression) 2) Then determine which plot of land has the larger area if x=3, how do you know? (2 points for answer, 2 points for explanation)

Answer 1

1) The area of a square is given by the formula A = s^2, where s represents the length of a side. In this case, the side length of the square plot of land is 4x^3, so the expression that represents the area of the square plot of land is (4x^3)^2.

Simplifying,
(4x^3)^2 = 4^2 * (x^3)^2 = 16 * x^6 = 16x^6.

The area of a rectangle is given by the formula A = l * w, where l represents the length and w represents the width. In this case, the length of the rectangular plot of land is 2x^5 and the width is 6x^2. So the expression that represents the area of the rectangular plot of land is (2x^5) * (6x^2).

Simplifying,
(2x^5) * (6x^2) = 2 * 6 * x^5 * x^2 = 12x^7.

2) To determine which plot of land has the larger area when x = 3, we can substitute x = 3 into the expressions we obtained in step 1 and compare the results.

For the square plot of land:
Area = 16(3)^6 = 16(729) = 11,664 square feet.

For the rectangular plot of land:
Area = 12(3)^7 = 12(2,187) = 26,244 square feet.

Therefore, the rectangular plot of land has the larger area when x = 3. The rectangular plot is larger by 26,244 - 11,664 = 14,580 square feet.

Explanation: We know that the area of a square is always less than or equal to the area of a rectangle with the same width and length. In this case, we can see that when x = 3, the value of the expression for the area of the rectangular plot is greater than the value of the expression for the area of the square plot. Thus, we conclude that the rectangular plot has the larger area.