1. A rhombus has a perimeter of 96, and the length of one of its diagonals is 32. The area of the circle inscribed in the rhombus can be expressed as k*pi/w where k and w are relatively prime positive integers. Find the value of k + w.
2. The base and height of an original rectangle are each increased by 15%. The area of the new rectangle is k/w times the area of the original rectangle. If k and w are relatively prime positive integers, find the value of k + w.
To solve the given problems, we will break down the steps involved in each question:
1. A rhombus has a perimeter of 96, and the length of one of its diagonals is 32. The area of the circle inscribed in the rhombus can be expressed as k * pi / w, where k and w are relatively prime positive integers. We need to find the value of k + w.
To find the area of the circle inscribed in the rhombus, we need to find the length of the side of the rhombus. Since a rhombus has all sides equal, the perimeter divided by 4 will give us the length of one side. In this case, the length of one side is 96/4 = 24.
Now, let's find the length of the other diagonal. Since the diagonals of a rhombus are perpendicular bisectors of each other, we can calculate the length of the other diagonal using the Pythagorean theorem.
Let's assume the length of the other diagonal is x. We know that one diagonal is 32, and the sides of the rhombus are 24. So, we have the following right-angled triangle:
x^2 = (24/2)^2 + 32^2
x^2 = 12^2 + 32^2
x^2 = 144 + 1024
x^2 = 1168
x = sqrt(1168)
x = 8sqrt(29)
Now, since the diagonals of a rhombus are perpendicular bisectors of each other, the radius of the inscribed circle is half the length of the diagonal. So, the radius of the circle is (8sqrt(29))/2 = 4sqrt(29).
Therefore, the area of the circle can be found using the formula: Area = pi * radius^2.
Area = pi * (4sqrt(29))^2
Area = 16pi * 29
Area = 464pi
Now, we know that the area of the circle can be expressed as k * pi / w, where k and w are relatively prime positive integers. In this case, k = 464 and w = 1.
Therefore, the value of k + w = 464 + 1 = 465.
Hence, the value of k + w is 465.
2. In the second problem, we have an original rectangle with a base and height. Both the base and height are increased by 15%, and the area of the new rectangle is k/w times the area of the original rectangle. We need to find the value of k + w.
Let's assume the original base of the rectangle is b and the original height is h. Then, the area of the original rectangle is b * h.
When both the base and height are increased by 15%, the new base becomes 1.15b, and the new height becomes 1.15h. Therefore, the area of the new rectangle is (1.15b) * (1.15h) = 1.3225 * b * h.
We are given that the area of the new rectangle is k/w times the area of the original rectangle. So, we have the following equation:
1.3225 * b * h = (k/w) * (b * h)
By comparing the coefficients, we can equate them:
1.3225 = k/w
To find the value of k + w, we need to determine two relatively prime positive integers whose ratio is 1.3225.
Under simplification, 1.3225 = (5305/4016).
Therefore, k = 5305 and w = 4016.
So, the value of k + w = 5305 + 4016 = 9321.
Hence, the value of k + w is 9321.