If w varies directly as r and inversely as the square of p, if w= 25 when r=5 and p=4, find r when w=16 and p=8
w = k r/p^2
25 = k (5)/16
k = 5*16 = 80
so
w = 80 r/p^2
16 = 80 r/64
r = 16*64/80 = 12.8
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To solve this problem, we need to use the formula for direct and inverse variation.
Let's assign the constants of variation as k1 and k2. Then, we can write the equation in terms of direct and inverse variation as:
w = k1 * r * (1/p^2)
Now, we can use the given information to find the values of k1 and k2.
When w = 25, r = 5, and p = 4, we can substitute these values into the equation:
25 = k1 * 5 * (1/4^2)
Simplifying, we have:
25 = k1 * 5 * (1/16)
To isolate k1, divide both sides of the equation by 5/16:
k1 = 25 / (5/16)
k1 = 25 * (16/5)
k1 = 80
Now that we have the value of k1, we can use it to find r when w = 16 and p = 8.
16 = 80 * r * (1/8^2)
Simplifying, we have:
16 = 80 * r * (1/64)
To isolate r, divide both sides of the equation by 80/64:
r = 16 / (80/64)
r = 16 * (64/80)
r = 12.8
Therefore, when w = 16 and p = 8, the value of r is 12.8.