QuestionJuly 24, 2025

Evaluate the following definite integral to two decimal places. int _(0)^30e^0.03te^0.05(30-t)dt int _(0)^30e^0.03te^0.05(30-t)dt=square (Round to two decimal places as needed.)

Evaluate the following definite integral to two decimal places. int _(0)^30e^0.03te^0.05(30-t)dt int _(0)^30e^0.03te^0.05(30-t)dt=square (Round to two decimal places as needed.)
Evaluate the following definite integral to two decimal places. int _(0)^30e^0.03te^0.05(30-t)dt int _(0)^30e^0.03te^0.05(30-t)dt=square (Round to two decimal places as needed.)

Solution
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Answer

336.98 Explanation 1. Simplify the integrand Combine exponents: e^{0.03t} \cdot e^{0.05(30-t)} = e^{0.03t + 0.05(30-t)} = e^{0.03t + 1.5 - 0.05t} = e^{1.5 - 0.02t}. 2. Set up the integral The integral becomes \int_{0}^{30} e^{1.5 - 0.02t} \, dt. 3. Integrate the function Use the formula for integrating exponential functions: \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C. Here, a = -0.02. The antiderivative is -\frac{1}{0.02} e^{1.5 - 0.02t} = -50 e^{1.5 - 0.02t}. 4. Evaluate the definite integral Calculate -50 e^{1.5 - 0.02t} from 0 to 30: At t = 30: -50 e^{1.5 - 0.02 \times 30} = -50 e^{0}. At t = 0: -50 e^{1.5 - 0.02 \times 0} = -50 e^{1.5}. Result: [-50 e^{0}] - [-50 e^{1.5}] = -50(1) + 50 e^{1.5}. 5. Compute the numerical value Calculate 50 e^{1.5} using a calculator: 50 \times e^{1.5} \approx 336.98.

Explanation

1. Simplify the integrand<br /> Combine exponents: $e^{0.03t} \cdot e^{0.05(30-t)} = e^{0.03t + 0.05(30-t)} = e^{0.03t + 1.5 - 0.05t} = e^{1.5 - 0.02t}$.<br /><br />2. Set up the integral<br /> The integral becomes $\int_{0}^{30} e^{1.5 - 0.02t} \, dt$.<br /><br />3. Integrate the function<br /> Use the formula for integrating exponential functions: $\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$. Here, $a = -0.02$.<br /> The antiderivative is $-\frac{1}{0.02} e^{1.5 - 0.02t} = -50 e^{1.5 - 0.02t}$.<br /><br />4. Evaluate the definite integral<br /> Calculate $-50 e^{1.5 - 0.02t}$ from $0$ to $30$:<br /> At $t = 30$: $-50 e^{1.5 - 0.02 \times 30} = -50 e^{0}$.<br /> At $t = 0$: $-50 e^{1.5 - 0.02 \times 0} = -50 e^{1.5}$.<br /> Result: $[-50 e^{0}] - [-50 e^{1.5}] = -50(1) + 50 e^{1.5}$.<br /><br />5. Compute the numerical value<br /> Calculate $50 e^{1.5}$ using a calculator: $50 \times e^{1.5} \approx 336.98$.
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