14. m varies inversely to n and directly to p. If m = 8 when n = 2 and p = 10, find p when y = 20 and n = 6.
=
m = kp/n
I have no idea where y fits in. But
mn/p is constant, so I think you want p such that
20*6/p = 8*2/10
To solve this problem, we need to use the inverse variation formula and the direct variation formula.
Inverse variation formula:
m = k/n
Direct variation formula:
m = kp
First, let's find the value of k.
When m = 8, n = 2, and p = 10:
8 = k/2
k = 16
Now, we can plug in the values of k, n, and m into the direct variation formula to solve for p when y = 20 and n = 6.
20 = 16p/6
To isolate p, we can cross multiply:
20 * 6 = 16p
120 = 16p
Divide both sides by 16 to solve for p:
120/16 = p
p = 7.5
Therefore, when y = 20 and n = 6, p = 7.5.
To solve this problem, we need to use the concept of inverse variation and direct variation.
Inverse variation means that when one variable increases, the other variable decreases in proportion. Mathematically, it can be represented as m = k/n, where k is the constant of variation.
Direct variation means that when one variable increases, the other variable also increases in proportion. Mathematically, it can be represented as m = kp, where k is the constant of variation.
From the given information, we know that m varies inversely with n and directly with p. Therefore, we can write the equation as m = k/n * kp.
To find the constant of variation k, we can substitute the initial values of m, n, and p into the equation. When m = 8, n = 2, and p = 10, we have:
8 = k/2 * 10
To solve for k, we can rearrange the equation as:
8 = 5k/2
Multiplying both sides by 2, we get:
16 = 5k
Dividing both sides by 5, we find:
k = 16/5
Now that we have the value of k, we can use it to find p when m = 20 and n = 6. Substituting these values into the equation, we get:
20 = (16/5)/6 * p
To solve for p, we can multiply both sides by 6 and divide by (16/5):
20 * 6 / (16/5) = p
Multiplying the numerator and denominator by 5 to simplify, we have:
120 / (16/5) = p
Dividing 120 by (16/5), we find:
p = 37.5
Therefore, when y = 20 and n = 6, the value of p is 37.5.