Question

P is partly constant and partly varies inversely as the square root of Q given that p=6when Q=9 and P=10 when Q=16 find P when Q=4

Answer 1

To find the relationship between \( P \) and \( Q \), given that \( P \) is partly constant and partly varies inversely as the square root of \( Q \), we can express \( P \) in the form:

\[
P = k + \frac{c}{\sqrt{Q}}
\]

where \( k \) is a constant and \( c \) is a coefficient that relates to the inverse variation with respect to the square root of \( Q \).

Given the conditions:

1. \( P = 6 \) when \( Q = 9 \)
2. \( P = 10 \) when \( Q = 16 \)

We can substitute these values into the equation to create a system of equations.

1. From \( P = 6 \) when \( Q = 9 \):

\[
6 = k + \frac{c}{\sqrt{9}} \implies 6 = k + \frac{c}{3}
\]

This can be rearranged to:

\[
k + \frac{c}{3} = 6 \quad \text{(Equation 1)}
\]

2. From \( P = 10 \) when \( Q = 16 \):

\[
10 = k + \frac{c}{\sqrt{16}} \implies 10 = k + \frac{c}{4}
\]

This can be rearranged to:

\[
k + \frac{c}{4} = 10 \quad \text{(Equation 2)}
\]

Now we have a system of equations from Equations 1 and 2:

\[
k + \frac{c}{3} = 6
\]
\[
k + \frac{c}{4} = 10
\]

Next, we can eliminate \( k \) by isolating it in both equations.

From Equation 1:

\[
k = 6 - \frac{c}{3}
\]

From Equation 2:

\[
k = 10 - \frac{c}{4}
\]

Setting the two expressions for \( k \) equal to each other:

\[
6 - \frac{c}{3} = 10 - \frac{c}{4}
\]

Now, we can solve for \( c \). First, multiply through by 12 to eliminate the fractions:

\[
12 \cdot 6 - 12 \cdot \frac{c}{3} = 12 \cdot 10 - 12 \cdot \frac{c}{4}
\]

This simplifies to:

\[
72 - 4c = 120 - 3c
\]

Now, rearranging gives:

\[
72 - 120 = c - 4c
\]

\[
-48 = -c
\]

Thus, we find:

\[
c = 48
\]

Now substitute \( c = 48 \) back into Equation 1 to find \( k \):

\[
k + \frac{48}{3} = 6
\]
\[
k + 16 = 6 \implies k = 6 - 16 = -10
\]

Now we have:

\[
k = -10 \quad \text{and} \quad c = 48
\]

Thus, the relationship between \( P \) and \( Q \) is:

\[
P = -10 + \frac{48}{\sqrt{Q}}
\]

Now to find \( P \) when \( Q = 4 \):

\[
P = -10 + \frac{48}{\sqrt{4}} = -10 + \frac{48}{2} = -10 + 24 = 14
\]

Therefore, the value of \( P \) when \( Q = 4 \) is:

\[
\boxed{14}
\]