Question

Write the exponential function corresponding to the given statement.

Answer 1

The exponential function corresponding to continuous growth with an initial value of 500 and a growth rate of 8% per year is y = 500 *
`e^(^0^.^0^8 ^*^ t^)`, where y represents the final value at time t.

Explanation:

In this scenario, the exponential function represents continuous growth, typically formulated as y = P *
`e^(^r^t^)`, where:

- P is the initial value,

- e is the base of the natural logarithm (approximately 2.718),

- r is the growth rate, and

- t is the time variable.

Given an initial value (P) of 500 and a growth rate (r) of 8% per year, we substitute these values into the general form. The growth rate needs to be converted to a decimal, so 8% becomes 0.08. Therefore, the function becomes y = 500 *
`e^(^0^.^0^8^ ^*t)`.

This function represents continuous growth because it uses the base of natural logarithm 'e.' As time (t) increases, the value of y grows continuously, compounding over time due to the exponential nature of the function. The variable t signifies the time at which we want to evaluate the function to find the corresponding value of y.

Understanding exponential functions is crucial, especially in scenarios of continuous growth or decay, as they depict how values change exponentially over time. In this context, the function y =
`500 * e^(^0^.^0^8^ *^ t^)`specifically models a situation of continuous growth with an initial value of 500 and an 8% annual growth rate.

Question:

Write the exponential function that represents continuous growth, where an initial value of 500 grows at a rate of 8% per year. Express this growth function in terms of y as a function of time (t), where y signifies the final value at time t. Use the formula y = P *
`e^(^r^t^)`, where P is the initial value, e is the base of the natural logarithm, r is the growth rate, and t is the time variable.