7. While on the road trip, Prolific's rental car's engine overheats, and he pulls over on the highway to allow it to cool. The outside temperature is 71°F. After 98 seconds, the temperature of the engine is 233°F. The temperature T, of the surface of a given engine after it has been cooling for t minutes can best be modeled by the function below, where T. is the temperature of the room and k is a constant. In (T-T.)=-kt +4.718 A. Compute the value of k to the nearest hundredth. B. Using this value of k, find the temperature T, of the engine that has been resting for a total of 212 seconds. Express your answer to the nearest degree. C. Engines operate safely between 190°F and 220°F. Determine if Prolific's car is safe to drive after 3 minutes of waiting.

7. While on the road trip, Prolific's rental car's engine overheats, and he pulls over on the
highway to allow it to cool. The outside temperature is 71°F. After 98 seconds, the temperature
of the engine is 233°F. The temperature T, of the surface of a given engine after it has been
cooling for t minutes can best be modeled by the function below, where T. is the temperature
of the room and k is a constant.
In (T-T.)=-kt +4.718

A. Compute the value of k to the nearest hundredth.

B. Using this value of k, find the temperature T, of the engine that has been resting for a
total of 212 seconds. Express your answer to the nearest degree.

C. Engines operate safely between 190°F and 220°F. Determine if Prolific's car is safe to
drive after 3 minutes of waiting.

Answer:

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Step-by-step explanation:

To compute the value of k, we can use the given information that after 98 seconds (t = 98), the temperature of the engine is 233°F (T = 233) with an outside temperature of 71°F (T₀ = 71). Plugging these values into the equation:

In (T - T₀) = -kt + 4.718

We have:

In (233 - 71) = -k(98) + 4.718

In (162) = -98k + 4.718

Taking the natural logarithm (ln) of both sides:

ln(162) = ln(-98k + 4.718)

Now, solve for k by rearranging the equation:

-98k + 4.718 = e^(ln(162))

-98k + 4.718 ≈ 5.2428 (rounded to four decimal places)

-98k ≈ 5.2428 - 4.718

-98k ≈ 0.5248

k ≈ 0.00535 (rounded to five decimal places)

A. The value of k, rounded to the nearest hundredth, is approximately 0.01.

To find the temperature T of the engine after resting for 212 seconds (t = 212), we can plug the values into the equation:

In (T - T₀) = -kt + 4.718

In (T - 71) = -(0.01)(212) + 4.718

In (T - 71) ≈ -2.12 + 4.718

In (T - 71) ≈ 2.598

Exponentiating both sides:

T - 71 ≈ e^(2.598)

T - 71 ≈ 13.4464

T ≈ 13.4464 + 71

T ≈ 84.4464

B. The temperature of the engine, after resting for a total of 212 seconds, is approximately 84°F.

To determine if the car is safe to drive after 3 minutes (t = 3 minutes = 180 seconds) of waiting, we can find the temperature T using the value of k:

In (T - 71) = -(0.01)(180) + 4.718

In (T - 71) ≈ -1.8 + 4.718

In (T - 71) ≈ 2.918

Exponentiating both sides:

T - 71 ≈ e^(2.918)

T - 71 ≈ 18.5277

T ≈ 18.5277 + 71

T ≈ 89.5277

The temperature of the engine after 3 minutes of waiting is approximately 90°F.

C. Since the temperature of the engine after 3 minutes of waiting is within the safe range of 190°F to 220°F, Prolific's car is safe to drive after 3 minutes of waiting.

A:

• To compute the value of k, we can use the given information that after 98 seconds (or 98/60 = 1.63 minutes), the temperature of the engine is 233°F.

• The outside temperature is 71°F. Plugging these values into the given equation ln(T - T.) = -kt + 4.718, we get ln(233 - 71) = -k * 1.63 + 4.718. Solving for k, we find that k ≈ 1.45 to the nearest hundredth.

B:

• Using the value of k = 1.45, we can find the temperature of the engine after it has been resting for a total of 212 seconds (or 212/60 = 3.53 minutes).

• Plugging these values into the equation ln(T - T.) = -kt + 4.718, we get ln(T - 71) = -1.45 * 3.53 + 4.718.

• Solving for T, we find that the temperature of the engine is approximately T ≈ 191°F to the nearest degree.

C:

• Since engines operate safely between 190°F and 220°F, and the temperature of Prolific’s car engine after resting for 3 minutes (or 180 seconds) is approximately 191°F, which falls within this range, it is safe to say that Prolific’s car is safe to drive after waiting for 3 minutes.

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