A stage has an area of 1,298 square feet and a perimeter of 162 feet. What are the dimensions of the stage?
Answers
Answer: The dimensions of the stage is 22 feet and 59 feet
Step-by-step explanation:
As we do not know if the stage is a rectangle or a square, we'll consider the stage to be a rectangle, as square is the special case of a rectangle.
Let's consider the dimensions to be X and Y.
Therefore,
X*Y = 1298 (area of rectangle is L * B) ------------------------------------------- (1)
2(X+Y) = 162 (Perimeter of rectangle is 2(L+B) ------------------------------- (2)
=> (X+Y) = 162/2 = 81 -------------------------------------------------------------------- (3)
Now we have the values of X+Y and X * Y, so we need to find the value of X-Y in order to find the value of X and Y
Squaring both the sides in (3)
Therefore,
(X+Y)² = 81²
X²+Y²+2XY = 81² [(a+b)² = a²+b²+2ab]
Subtracting 4XY both the sides,
Therefore,
X²+Y²+2XY-4XY = 81²-4XY
X²+Y²-2XY = 81²-4XY
(X-Y)² = 81²-4XY [(a-b)² = a²+b²-2ab]
(X-Y)² = 81² - 4(1298) [ substituting the value of XY ]
(X-Y)² = 6561 - 5192
(X-Y)² = 1369
Rooting both the sides,
Therefore,
X-Y = 37 --------------------------------------------------------------- (4)
Adding (3) and (4)
X+Y+X-Y = 37+81
2X = 118
X = 59
Substituting the value of X in (4)
Therefore,
59-Y = 37
Y = 59-37 = 22
The dimensions of the stage is 22 feet and 59 feet
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