1. Find the amount of time required for an investment to double at a rate of 12.3% if the interest is compounded continuously.

MY ANSWER: 5.635 yrs

2. Solve e^(0.2x) < 21.2 by using natural logarithms.

MY ANSWER: X<15.270

3. Convert log(base 5) 47 to a natural logarithm and evaluate.

My answer: 2.392

4. The pH of a water supply is 7.3. What is the concentration of hydrogen ions in the tested water?

I’m not sure but is it 5.012 x 10^-8

Check my answers?

All good!

1. Hmm, you're looking for the time it takes for an investment to double at a continuous interest rate of 12.3%? Well, I'm not great with numbers, but I can tell you that it sounds like it might take a while. Maybe longer than your average clown can hold their breath (which is about 5 seconds, by the way). So, I'm guessing it might be longer than 5.635 years. But hey, don't take my word for it, you may want to double-check your calculations.

2. Solve e^(0.2x) < 21.2 using natural logarithms? Well, let me tell you, that's like trying to find a needle in a haystack. Or in this case, finding "x" in a pile of mathematical noodles. My guess is that "x" should be less than 15.270, but remember, a clown's guesses are usually as reliable as a banana peel on a dance floor.

3. Converting log(base 5) 47 to a natural logarithm and evaluating? That's like asking a clown to tap dance while juggling chainsaws - it's a bit dangerous! But I'll give it a shot anyway. My rubber nose tells me that converting log(base 5) 47 to a natural logarithm gives you a number around 2.392. However, please remember to put on your math goggles before taking my answer too seriously!

4. Ah, the pH of a water supply. It's all about those hydrogen ions! Now, let me tell you a joke about hydrogen... just kidding! The concentration of hydrogen ions in the tested water with a pH of 7.3 is approximately 5.012 x 10^-8. But don't worry, they won't bite unless provoked by a rogue oxygen atom or a particularly mischievous clown.

1. To find the amount of time required for an investment to double at a rate of 12.3% with continuous compounding, you can use the formula:

t = ln(2) / (r)

where t is the time in years and r is the annual interest rate as a decimal. Plugging in the values, we have:

t = ln(2) / (0.123) ≈ 5.635 years.

So your answer is correct.

2. To solve the inequality e^(0.2x) < 21.2 using natural logarithms, you can take the natural logarithm of both sides of the inequality:

ln(e^(0.2x)) < ln(21.2)

0.2x < ln(21.2)

Now, divide both sides by 0.2:

x < ln(21.2)/0.2 ≈ 15.270

So your answer is correct.

3. To convert log(base 5) 47 to a natural logarithm, you can use the change of base formula:

log(base a) b = log(base c) b / log(base c) a

In this case, we want to change the base from 5 to e (natural logarithm). So we have:

log(base 5) 47 = log(base e) 47 / log(base e) 5

Using a calculator or by approximation, we have:

log(base e) 47 ≈ 2.392

So your answer is correct.

4. The pH of a water supply is given as 7.3. The concentration of hydrogen ions can be calculated using the formula:

[H+] = 10^(-pH)

Plugging in the value, we have:

[H+] = 10^(-7.3) ≈ 5.012 x 10^(-8)

So your answer is correct.

Overall, your answers are correct. Well done!

Sure, let's go through each question and check your answers.

1. To find the amount of time required for an investment to double at a rate of 12.3% with continuous compounding, you can use the formula:

t = ln(2) / (r)

where t is the time in years and r is the annual interest rate expressed as a decimal. Plugging in the values, we get:

t = ln(2) / (0.123)

Calculating this, we find t ≈ 5.635 years. Your answer is correct.

2. To solve the inequality e^(0.2x) < 21.2, we can take the natural logarithm (ln) on both sides of the inequality:

ln(e^(0.2x)) < ln(21.2)

0.2x < ln(21.2)

Then, we divide both sides of the inequality by 0.2:

x < ln(21.2) / 0.2

Evaluating this expression, we find x < 15.270. Your answer is correct.

3. To convert log(base 5) 47 to a natural logarithm, we can use the change of base formula:

log(base 5) 47 = ln(47) / ln(5)

Evaluating this expression, we find log(base 5) 47 ≈ 2.392. Your answer is correct.

4. The pH scale is used to measure the concentration of hydrogen ions (H+) in a solution. The formula to calculate pH is:

pH = -log[H+]

In this case, we have a pH of 7.3. To find the concentration of hydrogen ions ([H+]), we can rearrange the formula:

[H+] = 10^(-pH)

Plugging in the value, we get:

[H+] = 10^(-7.3)

Calculating this, we find [H+] ≈ 5.012 x 10^(-8). Your answer is correct.

Great job! All of your answers are correct.