Medical research indicates that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by

R=6e^12.77x
Where x is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident. What blood alcohol concentration corresponds to 20% risk of a car accident?

you are solving

6 e^12.77x = 20
e^12.77x = 10/3
ln both sides and use rules of logs

12.77x lne = ln(10/3) , recall that ln e = 1
x = ln(10/3) /12.77
= appr .094

To find the blood alcohol concentration that corresponds to a 20% risk of a car accident, we need to solve the equation "R = 6e^(12.77x)" when R is 20%.

First, let's convert the percent to a decimal by dividing by 100:
20% = 0.20

Substituting this value into the equation, we have:
0.20 = 6e^(12.77x)

To solve for x, we need to isolate the exponential term. Divide both sides of the equation by 6:
0.20/6 = e^(12.77x)

Simplifying further, we have:
0.0333 = e^(12.77x)

To solve for x, we need to take the natural logarithm (ln) of both sides of the equation:
ln(0.0333) = ln(e^(12.77x))

Using the property of logarithms, ln(e^a) = a, we can simplify the equation to:
ln(0.0333) = 12.77x

Now, divide both sides of the equation by 12.77:
ln(0.0333) / 12.77 = x

Using a calculator, we can calculate the value of x, which is approximately:
x ≈ -2.9508

Therefore, the blood alcohol concentration that corresponds to a 20% risk of a car accident is approximately -2.9508. However, since blood alcohol concentration can't be negative, we can assume that there is no blood alcohol concentration that corresponds to this specific risk level. It is important to note that the model used in this problem may not accurately represent real-world data and should be interpreted with caution.

To find the blood alcohol concentration (x) that corresponds to a 20% risk of a car accident (R), we can rearrange the equation and solve for x.

The given equation that models the risk of having a car accident is:
R = 6e^(12.77x)

First, let's convert the given risk value of 20% to decimal form by dividing it by 100:
20% = 0.20

Now, we can substitute the value of R into the equation:
0.20 = 6e^(12.77x)

To find x, we need to isolate the exponential term on the right-hand side of the equation. Let's divide both sides of the equation by 6:

0.20/6 = e^(12.77x)

Simplifying the left-hand side gives:

0.0333 = e^(12.77x)

To solve for x, we'll take the natural logarithm (ln) of both sides:

ln(0.0333) = ln(e^(12.77x))

Using the property of logarithms that ln(e^a) is equal to a, the equation becomes:

-3.4070 = 12.77x

Now, divide both sides of the equation by 12.77:

-3.4070/12.77 = x

Approximating the value gives:

x ≈ -0.267

Therefore, a blood alcohol concentration of approximately -0.267 corresponds to a 20% risk of a car accident. However, blood alcohol concentration cannot be negative, so this result is not physically meaningful.