Consider the following problem: The temperature of a soup is increasing at a rate of r(t)=30e^(-0.3 t) degrees Celsius per minute (where t is the time in minutes). At time t=0, the temperature of the soup is 21 degrees Celsius. By how much does the temperature increase between t=0 and t=8 minutes? Which expression can we use to solve the problem? Choose 1 answer: (A) r(8)-r(0) (B) int_(0)^(8)r(t)dt (c) r^(')(8)-r^(')(0) (D) int_(8)^(8)r(t)dt

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Consider the following problem: The temperature of a soup is increasing at a rate of  r(t)=30e^(-0.3 t) degrees Celsius per minute (where  t is the time in minutes). At time  t=0, the temperature of the soup is 21 degrees Celsius. By how much does the temperature increase between  t=0 and  t=8 minutes? Which expression can we use to solve the problem? Choose 1 answer: (A)  r(8)-r(0) (B)  int_(0)^(8)r(t)dt (c)  r^(')(8)-r^(')(0) (D)  int_(8)^(8)r(t)dt

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Rate of Temperature Change: To find the total increase in temperature over a period of time, we need to integrate the rate of temperature change over that time interval. The rate of temperature change is given by the function r(t) = 30e^(-0.3t). We are interested in the interval from t=0 to t=8 minutes.Integral Calculation: The correct expression to calculate the total increase in temperature from t=0 to t=8 is the integral of r(t) from 0 to 8. This is because the integral of a rate of change gives us the total change over the interval.Antiderivative Evaluation: The expression that represents the integral of r(t) from 0 to 8 is ∫(from 0 to 8) r(t) dt. This corresponds to choice (B) from the given options.Substitution and Simplification: To solve the problem, we calculate the integral of r(t) = 30e^(-0.3t) from t=0 to t=8. The integral of 30e^(-0.3t) with respect to t is -100e^(-0.3t), since the antiderivative of e^(-0.3t) is (-1/0.3)e^(-0.3t) and multiplying by 30 gives us -100e^(-0.3t).Final Calculation: We evaluate the antiderivative at the upper and lower limits of the integral and subtract the lower limit evaluation from the upper limit evaluation. This gives us: -100e^(-0.3*8) - (-100e^(-0.3*0))Substitute the values into the expression and simplify: -100e^(-2.4) - (-100e^(0))Since e^0 is 1, the expression simplifies to: -100e^(-2.4) - (-100 * 1)This further simplifies to: -100e^(-2.4) + 100Now we can calculate the numerical value of e^(-2.4) and then multiply by -100 and add 100 to find the total increase in temperature.Using a calculator, we find that e^(-2.4) is approximately 0.0907179533. Multiplying this by -100 gives us -9.07179533.Finally, we add 100 to -9.07179533 to get the total increase in temperature: -9.07179533 + 100 = 90.92820467

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