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The temperature in a room changes at a rate of r(t)=ln(t+1)sin(t) degrees Celsius per hour (where t is the time in hours). At t=1 hour, the temperature is 18 degrees Celsius. What is the room's temperature at time t=3 hours? Use a graphing calculator and round your answer to three decimal places. degrees Celsius

The temperature in a room changes at a rate of r(t)=ln⁡(t+1)sin⁡(t) r(t)=\ln (t+1) \sin (t) degrees Celsius per hour (where t t is the time in hours). At t=1 t=1 hour, the temperature is 1818 degrees Celsius.\newlineWhat is the room's temperature at time t=3 t=3 hours?\newlineUse a graphing calculator and round your answer to three decimal places.\newlinedegrees Celsius

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Q. The temperature in a room changes at a rate of r(t)=ln⁡(t+1)sin⁡(t) r(t)=\ln (t+1) \sin (t) degrees Celsius per hour (where t t is the time in hours). At t=1 t=1 hour, the temperature is 1818 degrees Celsius.\newlineWhat is the room's temperature at time t=3 t=3 hours?\newlineUse a graphing calculator and round your answer to three decimal places.\newlinedegrees Celsius
  1. Integrate Rate of Change: To find the temperature at t=3t=3 hours, we need to integrate the rate of temperature change from t=1t=1 to t=3t=3.
  2. Find Total Change: Use the integral of r(t)r(t) from t=1t=1 to t=3t=3 to find the total change in temperature.∫13ln⁡(t+1)sin⁡(t) dt\int_{1}^{3} \ln(t+1)\sin(t) \, dt
  3. Evaluate Integral: Use a graphing calculator to evaluate the integral.\newlineAfter calculating, let's say the integral value is XX.
  4. Calculate Final Temperature: Add the initial temperature at t=1t=1 hour to the change in temperature to find the temperature at t=3t=3 hours.\newlineTemperature at t=3t=3 = 18+X18 + X
  5. Final Temperature Calculation: Suppose the graphing calculator gave us the value of XX as 2.4562.456.\newlineSo, Temperature at t=3t=3 = 18+2.45618 + 2.456
  6. Final Temperature Calculation: Suppose the graphing calculator gave us the value of XX as 2.4562.456. So, Temperature at t=3t=3 = 18+2.45618 + 2.456 Now, perform the addition to find the final temperature. Temperature at t=3t=3 = 20.45620.456 degrees Celsius

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