How many hours did it take for the temperature to increase from 0 °C to 5 °C?

Steps to Solve

1. Set Up the Equations We need to find the times ( t ) when the temperature ( T(t) ) is equal to 0 °C and 5 °C.

First, set ( T(t) = 0 ): $$ \frac{75t^3 - 836t^2 + 3100t - 4185}{14t^2 + 10t - 35} = 0 $$

This implies the numerator must equal zero: $$ 75t^3 - 836t^2 + 3100t - 4185 = 0 $$

Next, set ( T(t) = 5 ): $$ \frac{75t^3 - 836t^2 + 3100t - 4185}{14t^2 + 10t - 35} = 5 $$

This leads to: $$ 75t^3 - 836t^2 + 3100t - 4185 - 5(14t^2 + 10t - 35) = 0 $$

2. Simplify the Second Equation Expanding the second equation gives: $$ 75t^3 - 836t^2 + 3100t - 4185 - 70t^2 - 50t + 175 = 0 $$

Combine like terms: $$ 75t^3 - 906t^2 + 3050t - 4008 = 0 $$

3. Solve the Polynomial Equations Now we have two polynomial equations:

• For ( T(t) = 0 ): $$ 75t^3 - 836t^2 + 3100t - 4185 = 0 $$
• For ( T(t) = 5 ): $$ 75t^3 - 906t^2 + 3050t - 4008 = 0 $$

You can use numerical methods or graphing methods to find the roots of these polynomials within the specified range ( 2 \leq t \leq 9 ).

4. Calculate the Time Difference Once you find the two valid ( t ) values from the equations, find the time difference: $$ \text{Time to reach 5 °C} - \text{Time to reach 0 °C} $$