Question

The area of a rectangular park is 8000m². If it had been 10m short and also 10m wider, the area would have increased by 100m². Calculate the original length and width of the park. a) Length: 80m, Width: 100m b) Length: 90m, Width: 110m c) Length: 70m, Width: 90m d) Length: 100m, Width: 120m

Answer 1

The original length and width of the rectangular park can be found by solving a system of equations. The equations are formed based on the given information about the area and the increase in area if the dimensions were changed. By solving the equations, the original length is found to be 80m and the original width is found to be 100m.

Step-by-step explanation:

Let the original length of the park be x and the original width be y. According to the given information, the area of the rectangular park is 8000m². Therefore, xy = 8000m². If the length is 10m shorter and the width is 10m wider, the area would have increased by 100m². This means that (x-10)(y+10) = 8100m².

Simplifying the equations, we have xy - 10y + 10x - 100 = 8100. Rearranging the terms, we get xy + 10x - 10y = 8200.

Now we can solve the two equations simultaneously to find the values of x and y. From the first equation, we have y = 8000/x. Substituting this value in the second equation, we get x(8000/x) + 10x - 10(8000/x) = 8200. Simplifying and rearranging the terms, we have 8000 + 10x² - 80000/x = 8200x. Multiplying through by x to eliminate the fraction, we get 10x³ - 8200x² + 80000x - 8000x = 0. Factoring out the common factor of 10x, we have 10x(x² - 820x + 8000 - 800) = 0. Simplifying further, we get 10x(x² - 820x + 7200) = 0.

Now we can factor the quadratic equation inside the parentheses. We are looking for two numbers that multiply to give 7200 and add up to -820. After trying different pairs of numbers, we find that -80 and -90 fit these criteria. Therefore, we can factor the equation as 10x(x - 80)(x - 90) = 0. This means that either x = 0, x - 80 = 0, or x - 90 = 0. Since the length cannot be 0, we discard x = 0 as a solution. Solving for x - 80 = 0 and x - 90 = 0, we find x = 80 and x = 90. Substituting these values back into the equation xy = 8000, we can find the corresponding values of y. For x = 80, y = 8000/80 = 100. For x = 90, y = 8000/90 = 88.89.

Therefore, the original length and width of the park are 80m and 100m, respectively.

Answer 2

The two options match the initial conditions:

a) Length: 80m, Width: 100m

d) Length: 100m, Width: 120m

Analyzing each option:

a) Length: 80m, Width: 100m:

Area: 80m * 100m = 8000m² (original area)

New area: (80m - 10m) * (100m + 10m) = 70m * 110m = 7700m²

Area increase: 8100m² - 8000m² = 100m²

This option matches the initial conditions.

b) Length: 90m, Width: 110m:

Area: 90m * 110m = 9900m²

New area: (90m - 10m) * (110m + 10m) = 80m * 120m = 9600m²

Area increase: 9700m² - 9900m² = -200m² (decrease instead of increase)

This option does not match the initial conditions.

c) Length: 70m, Width: 90m:

Area: 70m * 90m = 6300m²

New area: (70m - 10m) * (90m + 10m) = 60m * 100m = 6000m²

Area increase: 6400m² - 6300m² = 100m²

This option matches the initial conditions.

d) Length: 100m, Width: 120m:

Area: 100m * 120m = 12000m²

New area: (100m - 10m) * (120m + 10m) = 90m * 130m = 11700m²

Area increase: 12100m² - 12000m² = 100m²

This option matches the initial conditions.

Therefore, two options match the initial conditions:

a) Length: 80m, Width: 100m

d) Length: 100m, Width: 120m