Solve for a a=3500(1+(-0.24))^9.5
This question is asking you to compute the value of the variable 'a' using the provided formula: a = 3500(1 + (-0.24))^9.5. You must raise the sum inside the parentheses to the power of 9.5 and then multiply the result by 3500 to find the value of 'a'. This calculation involves exponential growth or decay depending upon the base number in the parentheses, which, in this case, is a negative growth rate because it is a negative number (-0.24).
$a = 3500 \left(\left(\right. 1 + \left(\right. - 0.24 \left.\right) \left.\right)\right)^{9.5}$
Answer
Solution:
Step 1:
Eliminate the brackets in the expression. $a = 3500(1 - 0.24)^{9.5}$
Step 2:
Simplify the expression $3500(1 - 0.24)^{9.5}$.
Step 2.1:
Calculate $1 - 0.24$. $a = 3500 \cdot (0.76)^{9.5}$
Step 2.2:
Compute the power of $0.76$ raised to $9.5$. $a = 3500 \cdot 0.07374441$
Step 2.3:
Multiply $3500$ by $0.07374441$. $a \approx 258.10545$
Knowledge Notes:
The problem involves solving for the variable $a$ in the exponential decay formula, which is a common mathematical operation in finance, physics, and other fields. The steps taken to solve the problem are as follows:
Exponential Decay Formula: The original expression $a = 3500(1 + (-0.24))^{9.5}$ represents an exponential decay model where the quantity decreases by a certain percentage over time or iterations.
Simplifying Expressions: Removing parentheses and simplifying expressions are basic algebraic skills required to solve equations. This involves understanding the order of operations and combining like terms.
Arithmetic Operations: The subtraction in step 2.1 and the exponentiation in step 2.2 are straightforward arithmetic operations. Subtraction is one of the four elementary arithmetic operations, and exponentiation is a mathematical operation involving the raising of a number to a power.
Exponentiation: Raising a number to a power, as in step 2.2, is a common operation in algebra. When dealing with exponential expressions, it's important to remember that the base is the number being multiplied by itself, and the exponent indicates how many times the base is used as a factor.
Multiplication: The final step involves multiplying two numbers, which is another basic arithmetic operation. Multiplication is a shortcut for repeated addition and is used extensively in various mathematical calculations.
Approximation: The final result is often rounded to a certain number of decimal places for practicality, especially when dealing with real-world applications. In this case, the result is rounded to five decimal places.
Use of Calculators: For steps involving exponentiation and multiplication with non-integer exponents, a calculator is typically used to obtain a numerical result.
The solution process involves a combination of algebraic simplification and arithmetic operations to solve for the variable $a$.
