175k views
5 votes
If the sum of the areas of the square and the rectangle is 94 cm square find x.

*perimeter of rectangla is 4x + 8

User Kean
by
7.8k points

1 Answer

3 votes

Answer:

Explanation:

Let the side of the square be 'a' and the dimensions of the rectangle be 'b' and 'c'.

We know that the area of a square = a^2

And the area of a rectangle = b*c

Given that the sum of the areas of the square and the rectangle is 94 cm^2, we can write the equation:

a^2 + b*c = 94

We are also given that the perimeter of the rectangle is 4x + 8. Since the perimeter of a rectangle is given by 2(b+c), we have:

2(b+c) = 4x + 8

b + c = 2x + 4

We need to find the value of 'x'.

We can express 'c' in terms of 'b' as c = 2x + 4 - b.

Substituting this value of 'c' in the equation a^2 + b*c = 94, we get:

a^2 + b(2x + 4 - b) = 94

Simplifying this equation, we get:

a^2 + 2bx - b^2 + 4b - 94 = 0

We know that the side of the square is equal to the length of the rectangle. Therefore, a = b.

Substituting this value of 'a' in the above equation, we get:

b^2 + 2bx + 4b - 94 = 0

We can now use the quadratic formula to solve for 'b':

b = (-2x ± sqrt(4x^2 - 4(4)(94))) / 2

b = (-2x ± sqrt(16x^2 - 1504)) / 2

b = -x ± sqrt(x^2 - 94)

Since the length of the rectangle cannot be negative, we take the positive value of 'b':

b = -x + sqrt(x^2 - 94)

We know that b + c = 2x + 4. Substituting the value of 'c' in terms of 'b', we get:

(-x + sqrt(x^2 - 94)) + (2x + 4 - (-x + sqrt(x^2 - 94))) = 2x + 4

Simplifying this equation, we get:

2sqrt(x^2 - 94) = 2

sqrt(x^2 - 94) =

User Pez Cuckow
by
8.3k points

No related questions found