The ratio of the areas of two squares is 32/63. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form (a*sqrt b)/c where a, b, and c are integers. What is the value of the sum a + b + c?
please post the full answer next time
To find the ratio of the side lengths of two squares, we need to take the square root of the ratio of their areas.
Given that the ratio of the areas is 32/63, let's find the square root of this ratio.
√(32/63)
To rationalize the denominator, we multiply both the numerator and denominator by the same number that will result in a rational number in the denominator. In this case, we need to multiply by √63 to rationalize the denominator.
√(32/63) * (√63/√63)
Simplifying,
= (√(32*63))/(√(63*63))
= (√(32*63))/(√(63^2))
= (√(32*63))/(63)
Factoring the square root,
= (√(2*16*63))/(63)
= (√(2*2*2*2*7*3*3))/(63)
= (√(2^4 * 7 * 3^2))/(63)
= (√(8*7*9))/(63)
= (√(56*9))/(63)
= (√(3*3*2*2*7))/(63)
= (3*2√(2*7))/(63)
= (6√14)/(63)
The ratio of the side lengths can be expressed as (6√14)/(63).
Now, let's find the values of a, b, and c.
From the expression, we have a = 6, b = 14, and c = 63.
Therefore, the sum a + b + c = 6 + 14 + 63 = 83.
To find the value of the sum a + b + c, we first need to determine the ratio of the side lengths of the two squares.
Let's assume the side length of the first square is x, and the side length of the second square is y.
According to the given information, the ratio of the areas of the two squares is 32/63. Since the area of a square is proportional to the square of its side length, we can write the equation:
(x^2) / (y^2) = 32/63
To rationalize the denominator, we need to multiply both sides of the equation by the conjugate of the denominator:
(x^2) / (y^2) * (c^2) / (c^2) = (32/63) * (c^2) / (c^2)
This simplifies to:
(x^2 * c^2) / (y^2 * c^2) = (32 * c^2) / (63 * c^2)
Now, we have a rationalized form:
(x * c) / (y * c) = (32 * c) / (63 * c)
Simplifying further, we get:
x / y = 32 / 63
Therefore, the ratio of the side lengths of the two squares is 32/63.
Now, we can express this ratio in the form (a*sqrt b)/c, where a, b, and c are integers.
To do this, we need to look for the largest perfect square that divides both 32 and 63. In this case, it's 1 because there are no perfect square factors that divide both numbers. Therefore, b = 1.
Now, let's try to simplify the ratio further:
32/63 = (a * sqrt(1)) / c
Since sqrt(1) = 1, the simplified form becomes:
32/63 = a / c
So, a = 32, and c = 63.
Finally, the sum a + b + c is:
32 + 1 + 63 = 96.
Hence, the value of the sum a + b + c is 96.
since the area scales by the square of the linear factor, the sides are in the ratio
√(32/63) = √(16*2 / 9*7) = 4/3 √(2/7)
I'll let you take it from there.