a rectangular plot of land measures [3x+9]m by [x-3]m and has an area of 648m^2

a) write an equation for the area of the plot in the form ax^2+bx+c=0
b) determine the dimensions of the plot.
c) another similar plot has an area of 2592m^2. find the dimensions of the plot

(a,b) (3x+9)(x-3) = 3(x+3)(x-3) = 3(x^2-9) = 648

x^2-9 = 216
x^2 = 225
x = 15
(c) 2592/648 = 4
so each dimension is √4=2 times as big

a) To write an equation for the area of the plot in the form ax^2 + bx + c = 0, we need to find the product of the lengths of the sides of the rectangle.

The given dimensions of the plot are: length = (3x + 9) m and width = (x - 3) m.

Area = length × width = (3x + 9) × (x - 3)

Expanding this expression, we get:
Area = 3x^2 - 9x + 27x - 81

Simplifying further, we have:
Area = 3x^2 + 18x - 81

Setting the area equal to 648 m^2, we get the equation:
3x^2 + 18x - 81 = 648

b) We can solve the equation 3x^2 + 18x - 81 = 648 to determine the dimensions of the plot.

Rearranging the equation to bring all terms to one side, we get:
3x^2 + 18x - 81 - 648 = 0

Simplifying, we have:
3x^2 + 18x - 729 = 0

To solve this quadratic equation, we can factor or use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = 18, and c = -729.

Using the quadratic formula, we have:
x = (-18 ± √(18^2 - 4 * 3 * -729)) / (2 * 3)
x = (-18 ± √(324 + 8748)) / 6
x = (-18 ± √9072) / 6
x = (-18 ± 95.2) / 6

Simplifying, we get:
x1 = (-18 + 95.2) / 6
x1 = 14.2

x2 = (-18 - 95.2) / 6
x2 = -18.2

We can ignore the negative value for x because it doesn't make sense in this context.

Therefore, the dimensions of the plot are:
Length = 3x + 9 = 3 * 14.2 + 9 = 42.6 + 9 = 51.6 m
Width = x - 3 = 14.2 - 3 = 11.2 m

c) We can apply the same steps to find the dimensions of the plot with an area of 2592 m^2.

Using the equation 3x^2 + 18x - 81 = 2592, we simplify it to:
3x^2 + 18x - 81 - 2592 = 0
3x^2 + 18x - 2673 = 0

Solving this quadratic equation using the quadratic formula, we get:
x = (-18 ± √(18^2 - 4 * 3 * -2673)) / (2 * 3)
x = (-18 ± √(324 + 32016)) / 6
x = (-18 ± √32340) / 6

Simplifying further:
x1 = (-18 + √32340) / 6
x1 = 14.95

x2 = (-18 - √32340) / 6
x2 ≈ -19.95

Ignoring the negative value, the dimensions of the plot are:
Length = 3x + 9 = 3 * 14.95 + 9 = 44.85 + 9 = 53.85 m
Width = x - 3 = 14.95 - 3 = 11.95 m

a) To find the equation for the area of the plot in the form ax^2 + bx + c = 0, we can use the formula for the area of a rectangle, which is length x width. In this case, the length is (3x + 9)m and the width is (x - 3)m. So, the equation becomes:

Area = (3x + 9)(x - 3) = 648m^2

b) Now, let's solve the equation to find the dimensions of the plot. We can start by expanding the equation:

3x^2 - 9x + 27x - 81 = 648

Simplifying it further:

3x^2 + 18x - 81 = 648

Subtracting 648 from both sides:

3x^2 + 18x - 729 = 0

Now, we can try to factorize or use the quadratic formula to solve for x. Unfortunately, this equation cannot be easily factorized, so let's use the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

For our equation, a = 3, b = 18, and c = -729. Substituting these values into the formula:

x = [-18 ± √(18^2 - 4(3)(-729))] / (2 * 3)

Calculating inside the square root:

x = [-18 ± √(324 + 8748)] / 6

x = [-18 ± √(9072)] / 6

x = [-18 ± 95.23] / 6

Now we solve for x:

1) x = (-18 + 95.23) / 6
x = 77.23 / 6
x ≈ 12.87

2) x = (-18 - 95.23) / 6
x = -113.23 / 6
x ≈ -18.87

As the dimensions of the plot cannot be negative, we can ignore the second solution.

So, the value of x is approximately 12.87.

Now we can substitute this value into the equations of the dimensions of the plot:
Length = 3x + 9 = 3(12.87) + 9 = 38.61m
Width = x - 3 = 12.87 - 3 = 9.87m

Therefore, the dimensions of the plot are approximately 38.61m by 9.87m.

c) To find the dimensions of another similar plot with an area of 2592m^2, we need to solve the equation:

Area = (3x + 9)(x - 3) = 2592m^2

Using the quadratic formula or factoring, we can find the value of x. Then, we can substitute this value back into the equations for length and width to find the dimensions of the second plot.