Question

The length of a rectangular plot of land is 4 miles greater than its width, and it’s area is 60 square miles. Find the dimensions of the plot of land

Answer 1

Width = 6 miles; length = 10 miles

Explanation:

We can determine the dimensions of the plot of land using a system of equations, where:

• L represents its length,
• and W represents its width.

First equation:

Since the plot's length is 4 miles greater than its width, our first equation is given by:

L = W + 4

Second equation:

Since the formula for the area of a rectangle is given by A = LW (i.e., area = length * width) and the area of the plot is 60 square miles, our second equation is given by:

60 = LW

Method to solve: Substitution:

Solving for W:

We can solve for W by substituting W + 4 for L in 60 = LW (i.e., the second equation:)

60 = (W + 4)(W)

60 = W^2 + 4W

Rewrite the equation in the standard form of a quadratic (i.e., ax^2 + bx + c = 0):

Now, we can arrange the equation to be in the standard form of a quadratic:

(60 = W^2 + 4W) - W^2 - 4W

-W^2 - 4W + 60 = 0

Note that -W^2 is the same as -1W^2, meaning we can write our equation as:

-1W^2 - 4W + 60 = 0

Solving the quadratic with the quadratic formula:

The quadratic formula is given by:

`x=(-b+/-√(b^2-4ac) )/(2a)`, where:

• x represent the solution(s) to the quadratic,
• and a, b, and c are numbers from the standard form.

Thus, we can solve the quadratic by substituting -1, -4, and 60 for a, b, and c in the quadratic formula:

`w=(-(-4)+/-√((-4)^2-4(-1)(60)) )/(2(-1))\\ \\w=(4+/-√(256) )/(-2)`

Now, we can begin splitting up the quadratic to find the positive solution and the negative solution:

Positive solution:

`w=(4+√(256) )/(-2)\\ \\w=(4)/(-2)+(√(256) )/(-2)\\ \\ w=-2+(16)/(-2)\\ \\ w=-2-8\\\\w=-10`

• Therefore, the positive solution is -10. Since width can't be negative, we can eliminate this answer, meaning that the width will be our negative solution.

Negative solution:

`w=(4-√(256) )/(-2)\\\\w=(4)/(-2)-(√(256) )/(-2)\\ \\w=-2-(16)/(-2)\\ \\w=-2-(-8)\\\\w=-2+8\\\\w=6`

Therefore, the width of the plot is 6 miles.

Solving for L:

Now, we can solve for L by plugging in 6 for W in L = W + 4:

L = 6 + 4

L = 10

Therefore, the length of the plot is 10 miles.