Gavin is subdividing land into two plots, where one plot is in the shape of a square and the other plot is in the shape of a rectangle. The square plot of land has a side length of 6x3 feet, and the rectangular plot of land has a length of 3x5 feet and a width of 7x2 feet. Use the properties of exponents to determine the expression that represents the area for each plot of land. Then analyze which plot of land has a larger area if x is 3.
Well, well, well! Looks like Gavin is playing with his lands! Let's do some math and determine the areas.
For the square plot of land, we know that the side length is 6x3 feet. So, to find the area of a square, we need to multiply the side length by itself. Therefore, the expression for the area of the square plot of land is (6x3) * (6x3).
For the rectangular plot of land, we know the length is 3x5 feet and the width is 7x2 feet. To find the area of a rectangle, we need to multiply the length by the width. So, the expression for the area of the rectangular plot of land is (3x5) * (7x2).
Now, let's evaluate the expressions for x = 3.
For the square plot of land: (6x3) * (6x3) = 18 * 18 = 324 square feet.
For the rectangular plot of land: (3x5) * (7x2) = 15 * 14 = 210 square feet.
So, when x is equal to 3, the square plot of land has a larger area with 324 square feet, while the rectangular plot of land has 210 square feet.
To find the area of a square, you need to multiply the length of one side by itself. The expression that represents the area of the square plot of land with a side length of 6x3 feet is:
Area of the square = (6x3)^2 = 6x3 * 6x3 = 36x^2 * 9 = 324x^2 square feet
To find the area of a rectangle, you need to multiply the length by the width. The expression that represents the area of the rectangular plot of land with a length of 3x5 feet and a width of 7x2 feet is:
Area of the rectangle = (3x5) * (7x2) = 15x * 14x = 210x^2 square feet
Now, we need to analyze which plot of land has a larger area when x is 3. We can substitute x = 3 into the expressions we found for the areas:
For the square plot:
Area = 324x^2
Area = 324(3)^2
Area = 324(9)
Area = 2916 square feet
For the rectangular plot:
Area = 210x^2
Area = 210(3)^2
Area = 210(9)
Area = 1890 square feet
Therefore, when x is 3, the square plot of land has a larger area with 2916 square feet, compared to the rectangular plot which has an area of 1890 square feet.
To determine the expression that represents the area for each plot of land, we need to use the formula for area.
The area of a square is given by the formula: side length squared.
Since the side length of the square plot is 6x3 feet, the expression for its area is (6x3)^2.
To simplify this expression, we can apply the properties of exponents. When raising a power to a power, we multiply the exponents. In this case, (6x3)^2 becomes (6^2) * (x^3)^2, which simplifies to 36 * x^6.
So, the expression representing the area of the square plot is 36x^6 square feet.
The area of a rectangle is given by the formula: length multiplied by width.
For the rectangular plot of land, the length is 3x5 feet and the width is 7x2 feet. So, the expression for its area is (3x5) * (7x2).
To simplify this expression, we can again apply the properties of exponents. Multiplying the expressions (3x5) and (7x2), we get (3 * 7) * (x^5 * x^2), which simplifies to 21 * x^7.
So, the expression representing the area of the rectangular plot is 21x^7 square feet.
To analyze which plot of land has a larger area when x is 3, we substitute x = 3 into both expressions and compare the results.
For the square plot: 36 * 3^6 = 36 * 729 = 26244 square feet.
For the rectangular plot: 21 * 3^7 = 21 * 2187 = 45867 square feet.
Therefore, when x is 3, the rectangular plot of land has a larger area.
(x^3)(x^3) = x^(3+3) = x^6
similarly for the rectangle.
now work your magic.