Question

A rectangular parcel of land has an area of 3,000 ft2. A diagonal between opposite corners is measured to be 10 ft longer than one side of the parcel. What are the dimensions of the land, correct to the nearest foot? ft (smaller value) by ft (larger value)

Answer 1

Answer:

  40 ft by 75 ft

Step-by-step explanation:

Let x and y represent the dimensions, with x being the dimension 10 less than the diagonal. The Pythagorean theorem tells us ...

  (x +10)² = x² + y²

  x² +20x +100 = x² +y²

Then solving for x gives ...

  x = (y² -100)/20

and substituting for x in the area formula, we get ...

  xy = 3000

  ((y² -100)/20)y = 3000

  y³ -100y = 60000

This cubic can be solved a variety of ways. One is "guess and check". The solution will be very near ∛60000 ≈ 39, so we might guess y=40. Checking, we see that is indeed the solution:

  40³ -100·40 = 60000

  64000 -4000 = 60000 . . . . verifies y=40 is the solution

Then x=3000/40 = 75, and we have found the dimensions of the rectangle to be 40 ft by 75 ft.

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Another way to solve the cubic is graphically. We can rewrite it as ...

  y³ -100y -60000 = 0

and look for the zero of the cubic expression on the left. A graph shows it to be y=40.

Answer 2

Answer:

40 ft × 75 ft

Step-by-step explanation:

Let x be the one side ( in feet ) of the rectangular parcel,

So, the diagonal between opposite corners = ( x + 10 ) feet,

Let y be the other side of the rectangle ( adjacent to side x ),

The area of the rectangle,

A = x × y,

According to the question,

A = 3000 square ft,

$\implies xy = 3000\implies y=\frac{3000}{x}$

∵ In a rectangle,

$D^2=a^2+b^2$

Where, a and b are the adjacent side of the rectangle and D is the diagonal,

$(x+10)^2 = x^2 + y^2$

$(x+10)^2 = x^2 + \frac{9000000}{x^2}$

$x^2(x+10)^2 = x^4 + 9000000$

$x^2(x^2+100+20x) = x^4 + 9000000$

$x^4+100x^2 + 20x^3 = x^4 + 9000000$

$20x^3 + 100x^2 - 9000000=0$

$x^3 + 5x^2 - 450000 = 0$

By graphing the equation,

We found that,

The only real zeros of the equation is at x = 75,

Hence, the one side of the rectangle = 75 ft,

And, second side = $\frac{3000}{75}$ = 40 ft

Hence, the dimension of the land is 40 ft × 75 ft