The function f is defined by the given equation. The equation can be rewritten as f(x) = (1 + p/100)^x, where p is a constant. Which of the following is closest to the value of p?

Steps to Solve

1. Identify the given function forms
   The equation is given in two forms:
   $$ f(x) = (1.84)^{\frac{x}{4}} $$
   and
   $$ f(x) = \left(1 + \frac{p}{100}\right)^{x} $$

2. Rewrite the first function
   To match the forms, we can express the exponential function: $$ (1.84)^{\frac{x}{4}} = \left((1.84)^{\frac{1}{4}}\right)^{x} $$
   This means that the base is: $$ 1 + \frac{p}{100} = (1.84)^{\frac{1}{4}} $$

3. Calculate $(1.84)^{\frac{1}{4}}$
   Now let's compute the value of $(1.84)^{\frac{1}{4}}$:
   Using a calculator or computer, we find: $$ (1.84)^{\frac{1}{4}} \approx 1.1803 $$

4. Set up the equation for p
   Now we can set up the equation as follows:
   $$ 1 + \frac{p}{100} = 1.1803 $$

5. Solve for p
   Subtract 1 from both sides: $$ \frac{p}{100} = 1.1803 - 1 = 0.1803 $$
   Now multiply both sides by 100 to find p: $$ p = 0.1803 \times 100 \approx 18.03 $$

6. Identify the closest option
   We check the options provided; the closest value to 18.03 is:

• 16
• 21
• 46
• 96

Therefore, the answer is 21 since it is the nearest to 18.03.