Question

If v varies partly as aq and partly as q square when v=34,q=2 and v=190
,q=5find vwhen q=10

Answer 1

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:

1. When \( v = 34 \) and \( q = 2 \):
\[
34 = k_1 \cdot a \cdot 2 + k_2 \cdot 2^2
\]
\[
34 = 2a k_1 + 4k_2
\]

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:

1. \( 2a k_1 + 4k_2 = 34 \) \quad (1)
2. \( 5a k_1 + 25k_2 = 190 \) \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 \( \boxed{703} \).