Question:

A chemical reaction involving a liquid compound is studied in a laboratory. It is found that the volume $V$ of the compound, measured in milliliters, is related to the temperature $T$, measured in degrees Celsius, by the equation $V = 5T^2 + 10T + 100$, where $T$ ranges from 0 to 10 degrees Celsius.

(a) Use linear approximation to estimate the change in volume when the temperature increases from $T = 3$ degrees Celsius to $T = 3.1$ degrees Celsius.

(b) Use differentials to approximate the change in volume when the temperature increases from $T = 9$ degrees Celsius to $T = 9.05$ degrees Celsius.

Answer:

(a) To estimate the change in volume using linear approximation, we need to find the derivative of the volume function $V$ with respect to temperature $T$.

The derivative of $V$ with respect to $T$ can be found by differentiating each term using the power rule:

$\frac{dV}{dT} = \frac{d}{dT} (5T^2 + 10T + 100) = 10T + 10$

Now, we can find the linear approximation using the formula:

$\Delta V \approx \frac{dV}{dT} \Delta T$

Given that the temperature change $\Delta T = (3.1 - 3)$ degrees Celsius, we substitute the values:

$\Delta V \approx (10T + 10) \Delta T$

$\Delta V \approx (10\cdot3 + 10) \cdot (0.1)$

$\Delta V \approx (30 + 10) \cdot (0.1)$

$\Delta V \approx 4$ milliliters

Therefore, the volume of the compound is estimated to increase by approximately 4 milliliters when the temperature increases from 3 degrees Celsius to 3.1 degrees Celsius.

(b) To approximate the change in volume using differentials, we can use the formula:

$\Delta V \approx \frac{dV}{dT} \Delta T$

We already found the derivative of $\frac{dV}{dT} = 10T + 10$.

Given that the temperature change $\Delta T = (9.05 - 9)$ degrees Celsius, we substitute the values:

$\Delta V \approx (10T + 10) \Delta T$

$\Delta V \approx (10\cdot9 + 10) \cdot (0.05)$

$\Delta V \approx (90 + 10) \cdot (0.05)$

$\Delta V \approx 5$ milliliters

Therefore, the volume of the compound is approximated to increase by approximately 5 milliliters when the temperature increases from 9 degrees Celsius to 9.05 degrees Celsius.