Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Find the function values. \(A(P, r, t)=P e^{r t}\) (a) \(A(500,0.10,5)\) (b) \(A(1500,0.12,20)\)

Step-by-Step Solution

Answer

The function value for the first part (a) comes out to be approximately 824.36 depending on the rounding and for the second part (b), it is approximately 10863.67 based on the provided inputs.

Step 1: Apply the compound interest formula for the first problem

Plug in the given values of P = 500, r = 0.10 and t = 5 into the formula \(A = P * e^{r*t}\) which turns into A = 500 * e ^ (0.10 * 5)

Step 2: Solve the first problem

Calculate e to the power of (0.10 * 5) and then multiply the result with 500. It's crucial to perform the exponent operation first in accordance with the order of operations (BIDMAS/BODMAS).

Step 3: Apply the compound interest formula for the second problem

Same as in step 1, but this time using the values P = 1500, r = 0.12 and t = 20. The formula then turns into A = 1500 * e ^ (0.12*20).

Step 4: Solve the second problem

Calculate e to the power of (0.12 * 20) and then multiply the result with 1500. As explained earlier, remember to perform the exponent operation first following the order of operations.

Key Concepts

Exponential Functions

Understanding the concept of exponential functions is crucial when dealing with compound interest problems. An exponential function can be represented as f(x) = a * e^{bx}, where e is the base of the natural logarithm, approximately equal to 2.71828, and a and b are constants. These functions grow at a rate proportional to their value, which is why they're often used to model growth and decay in real-world situations, like population growth or radioactive decay.

In the context of our compound interest formula, A(P, r, t) = Pe^{rt}, the amount A is the function value we want to find, where P represents the principal amount, r is the interest rate, and t is time in years. The presence of the exponential function in this formula shows how investment grows over time at a rate that increases continuously, a concept that is part of the magic behind compound interest.

When the values for P, r, and t are substituted into the formula, the value of A that we obtain reflects the future value of an investment after earning continuous interest for t years at a rate of r.

Order of Operations

Order of operations is an essential principle in mathematics to avoid ambiguity in mathematical expressions. BIDMAS/BODMAS is an acronym to help remember the order: Brackets, Indices (Powers and roots, such as squares and square roots), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right).

In our compound interest formula, A = Pe^{rt}, the exponent rt must be calculated first before any multiplication. This is because indices (exponents) come before multiplication in the order of operations. If we were to multiply P by e before raising it to the power of rt, we would get a completely different, and incorrect, result. A proper application of the order of operations ensures the accuracy of compound interest calculations and is vital for solving various types of mathematical problems.

Mathematical Problem Solving

Mathematical problem solving involves understanding the problem, devising a plan, carrying out the plan, and looking back to check and interpret the results. When solving compound interest problems with the formula A(P, r, t) = Pe^{rt}, it's important to carefully identify P (principal), r (rate), and t (time) provided in the problem.

Once the values are established, the next step is to apply the formula correctly, which includes adhering to the order of operations. This process requires attention to detail to ensure each step is correctly followed. After performing the calculations, verifying the results to ensure they make sense in the context of the problem helps solidify understanding and completes the problem-solving process.

In the given examples, we apply the formula to different sets of values to find the future value of investments. This disciplined approach to solving mathematics problems can be applied broadly across various topics, enhancing students’ overall mathematical proficiency.