A Can Of Soda At 83 °F Is Placed In A Refrigerator

13 A Can Of Soda At 83 F Is Placed In A Refrigerator That Maintains

A can of soda at 83° F is placed in a refrigerator that maintains a constant temperature of 35° F. The temperature T (t) of the soda t minutes after it is placed in the refrigerator is given by T(t) = 35 + 48 e^(-0.058 t). Find the temperature of the soda 15 minutes after it is placed in the refrigerator. Round to the nearest tenth of a degree.

Suppose $5,700 is invested in an account at an annual interest rate of 6.8% compounded continuously. How long (to the nearest tenth of a year) will it take for the investment to double in size?

Multiply and simplify: (9 - 2i)(3 + 4i). Write the answer in the form a + bi, where a and b are real numbers.

Paper For Above instruction

Introduction

The study of temperature change, continuous investment growth, and complex number multiplication are fundamental aspects of mathematics applicable across numerous fields. Understanding how to model temperature decay with exponential functions, calculating the time for investment growth using logarithms, and simplifying complex numbers are essential skills in applied mathematics. This paper explores each of these topics, providing detailed solutions and insights into their mathematical principles.

Temperature Decay in a Refrigerator

The problem involves modeling the cooling of a soda can using Newton's Law of Cooling, expressed as T(t) = T_env + (T_initial - T_env)e^(-kt). Given T(t) = 35 + 48e^(-0.058 t), where T_initial = 83°F and T_env = 35°F, we are tasked with calculating the soda's temperature after 15 minutes.

Substituting t = 15 into the function: T(15) = 35 + 48e^(-0.058 15). Calculating the exponent: -0.058 15 = -0.87. Then, e^(-0.87) ≈ 0.419. Multiplying by 48: 48 * 0.419 ≈ 20.112. Finally, add 35 to find T(15): 35 + 20.112 ≈ 55.1°F. Therefore, the temperature of the soda 15 minutes after being placed in the refrigerator is approximately 55.1°F.

Continuous Investment Doubling Time

The investment grows according to the formula A = Pe^{rt}, where P = 5700, r = 0.068, and A = 2P = 11400. To find the time to double, set up the equation: 11400 = 5700 e^{0.068 t}. Dividing both sides by 5700: 2 = e^{0.068 t}. Taking natural logarithms: ln(2) = 0.068 t. Solving for t: t = ln(2) / 0.068 ≈ 0.6931 / 0.068 ≈ 10.2 years. Hence, it takes approximately 10.2 years for the investment to double.

Multiplication and Simplification of Complex Numbers

Given the complex numbers (9 - 2i) and (3 + 4i), their product is computed as follows:

(9 - 2i)(3 + 4i) = 9 3 + 9 4i - 2i 3 - 2i 4i = 27 + 36i - 6i - 8i^2. Since i^2 = -1, substitute: -8(-1) = +8.

Combine like terms: (27 + 8) + (36i - 6i) = 35 + 30i. Thus, the simplified form in a + bi format is 35 + 30i.

Conclusion

Through these problems, we observe the application of exponential decay models, logarithmic calculations for compound interest, and algebraic manipulation of complex numbers. Mastery of these concepts facilitates a deeper understanding of both theoretical and real-world mathematical applications, highlighting their importance across scientific and financial disciplines.

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