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,
[tex]\implies xy = 3000\implies y=\frac{3000}{x}[/tex]
∵ In a rectangle,
[tex]D^2=a^2+b^2[/tex]
Where, a and b are the adjacent side of the rectangle and D is the diagonal,
[tex](x+10)^2 = x^2 + y^2[/tex]
[tex](x+10)^2 = x^2 + \frac{9000000}{x^2}[/tex]
[tex]x^2(x+10)^2 = x^4 + 9000000[/tex]
[tex]x^2(x^2+100+20x) = x^4 + 9000000[/tex]
[tex]x^4+100x^2 + 20x^3 = x^4 + 9000000[/tex]
[tex]20x^3 + 100x^2 - 9000000=0[/tex]
[tex]x^3 + 5x^2 - 450000 = 0[/tex]
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 = [tex]\frac{3000}{75}[/tex] = 40 ft
Hence, the dimension of the land is 40 ft × 75 ft
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.
