To find the relationship between \( v \) and \( q \), we start with the concept that \( v \) varies partly as \( aq \) and partly as \( q^2 \). This can be expressed mathematically as:
\[ v = k_1 \cdot aq + k_2 \cdot q^2 \]
where \( k_1 \) and \( k_2 \) are constants that we need to determine.
Step 1: Set up equations from given values
From the information you provided, we have two scenarios:
-
When \( v = 34 \) and \( q = 2 \): \[ 34 = k_1 \cdot a \cdot 2 + k_2 \cdot 2^2 \] \[ 34 = 2a k_1 + 4k_2 \]
-
When \( v = 190 \) and \( q = 5 \): \[ 190 = k_1 \cdot a \cdot 5 + k_2 \cdot 5^2 \] \[ 190 = 5a k_1 + 25k_2 \]
Step 2: Write the system of equations
We now have a system of equations:
- \( 2a k_1 + 4k_2 = 34 \) \quad (1)
- \( 5a k_1 + 25k_2 = 190 \) \quad (2)
Step 3: Solve the system of equations
We can manipulate equation (1) and (2) to solve for \( k_1 \) and \( k_2 \).
From equation (1), we can express \( k_2 \) in terms of \( a \) and \( k_1 \):
\[ k_2 = \frac{34 - 2ak_1}{4} \]
Substituting \( k_2 \) into equation (2):
\[ 5ak_1 + 25\left(\frac{34 - 2ak_1}{4}\right) = 190 \]
Multiply through by 4 to eliminate the fraction:
\[ 20ak_1 + 25(34 - 2ak_1) = 760 \]
Expanding this gives:
\[ 20ak_1 + 850 - 50ak_1 = 760 \]
Combining the terms:
\[ -30ak_1 + 850 = 760 \]
Rearranging yields:
\[ -30ak_1 = 760 - 850 \]
\[ -30ak_1 = -90 \]
\[ 30ak_1 = 90 \]
\[ ak_1 = 3 \quad \Rightarrow \quad k_1 = \frac{3}{a} \]
Step 4: Substitute \( k_1 \) back to find \( k_2 \)
Substitute \( k_1 \) back into equation (1):
\[ 2a\left(\frac{3}{a}\right) + 4k_2 = 34 \]
\[ 6 + 4k_2 = 34 \]
\[ 4k_2 = 28 \]
\[ k_2 = 7 \]
Step 5: Final equation for \( v \)
Now substitute \( k_1 \) and \( k_2 \) back into the original equation:
\[ v = \frac{3}{a} \cdot aq + 7q^2 \]
\[ v = 3 + 7q^2 \]
Step 6: Find \( v \) when \( q = 10 \)
Now we substitute \( q = 10 \) into our final equation:
\[ v = 3 + 7(10^2) \]
\[ v = 3 + 7 \cdot 100 \]
\[ v = 3 + 700 \]
\[ v = 703 \]
Thus, the value of \( v \) when \( q = 10 \) is \( \boxed{703} \).