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.
A2/A1 = (1.15)^2 = 1.3225 = k/w
If k and w must be integers, and "relatively prime", then
k/w = 13225/10000 must be reduced to the lowest ratio of prime numbers
13225/10000 = 529/400
Are you asking for the value of k+ w, or k AND w? It looks like 529 and 400 are your k and w.
Actually, 529 is not a prime number; it is 23^2. I assume what is really being asked for is the lowest ratio of integers. That would be 529/400. So k would be 529 and w would be 400.
To solve this problem, let's first find the area of the original rectangle and the new rectangle.
Let the original base of the rectangle be b and the original height be h. Then, the area of the original rectangle is A = b * h.
After increasing the base and height by 15%, the new base of the rectangle becomes 1.15b (increased by 15%) and the new height becomes 1.15h (increased by 15%). Therefore, the area of the new rectangle is A' = (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 equation:
A' = (k/w) * A
Substituting the values we found for A and A', we get:
1.3225 * b * h = (k/w) * (b * h)
Cancellation of b and h gives:
1.3225 = k/w
Since k and w are relatively prime positive integers, this means that the fraction k/w is in simplest form. Therefore, k = 1.3225 and w = 1.
To find the value of k + w, we add the values together:
k + w = 1.3225 + 1 = 2.3225
So, the value of k + w is 2.3225.