If `t(u)=au^(2)-3u+1 and t(3)=37`, what is a?

Answer

To solve the problem, we need to find the value of \( a \) in the function \( t(u) = au^2 - 3u + 1 \) given that \( t(3) = 37 \).

### Step-by-Step Solution:

1. **Substitute \( u = 3 \) into the function**:
\[ t(3) = a(3^2) - 3(3) + 1 \]

2. **Calculate \( 3^2 \)**:
\[ 3^2 = 9 \]
So, we can rewrite the equation as:
\[ t(3) = a(9) - 3(3) + 1 \]

3. **Calculate \( -3(3) \)**:
\[ -3(3) = -9 \]
Now, substitute this back into the equation:
\[ t(3) = 9a - 9 + 1 \]

4. **Combine like terms**:
\[ t(3) = 9a - 8 \]

5. **Set the equation equal to 37** (since \( t(3) = 37 \)):
\[ 9a - 8 = 37 \]

6. **Add 8 to both sides**:
\[ 9a = 37 + 8 \]
\[ 9a = 45 \]

7. **Divide both sides by 9**:
\[ a = \frac{45}{9} \]
\[ a = 5 \]

### Final Answer:
The value of \( a \) is \( 5 \).