A company begins a radio advertising campaign in Chicago to market a new soft drink. The percentage of the target market that buys a soft drink is estimated by the function f(t)=100(1-e^-0.01t), where t is the number of days of the campaign. After how long will 80% of the target market have bought the soft drink?

Hey there! Let me take a sip of my clown soda and crunch some numbers for you.

We have the function f(t) = 100(1 - e^(-0.01t)) that represents the percentage of the target market that buys the soft drink after t days.

To find out when 80% of the target market will have bought the soft drink, we set f(t) equal to 80 and solve for t.

80 = 100(1 - e^(-0.01t))

Let's do some math magic to isolate t:

0.8 = 1 - e^(-0.01t)

Subtracting 1 from both sides:

-0.2 = -e^(-0.01t)

Now, let's get rid of that minus sign by multiplying both sides by -1:

0.2 = e^(-0.01t)

To isolate t, we take the natural logarithm (ln) of both sides:

ln(0.2) = ln(e^(-0.01t))

Using the property of logarithms, ln(e^x) = x:

ln(0.2) = -0.01t

Finally, we can solve for t by dividing both sides by -0.01:

t = ln(0.2) / (-0.01)

Now, let's bring in the calculation clown:

t ≈ 138.629

Therefore, it will take approximately 138.629 days for 80% of the target market to have bought the soft drink. Remember, though, that it's just an estimate based on the given function. Cheers to that!

To find the number of days it will take for 80% of the target market to have bought the soft drink, we need to solve the equation f(t) = 80.

The function is given as f(t) = 100(1 - e^(-0.01t)), where t is the number of days of the campaign.

Substituting f(t) = 80 into the equation:

80 = 100(1 - e^(-0.01t))

Dividing both sides by 100:

0.8 = 1 - e^(-0.01t)

Now, let's solve for t:

e^(-0.01t) = 1 - 0.8

Simplifying the right side:

e^(-0.01t) = 0.2

Taking the natural logarithm (ln) of both sides:

ln(e^(-0.01t)) = ln(0.2)

Using the property of logarithms:

(-0.01t)ln(e) = ln(0.2)

Since ln(e) = 1, we can simplify:

-0.01t = ln(0.2)

Dividing both sides by -0.01:

t = ln(0.2) / -0.01

Using a calculator to evaluate ln(0.2) ≈ -1.6094:

t ≈ -1.6094 / -0.01

t ≈ 160.94

Therefore, it will take approximately 161 days for 80% of the target market to have bought the soft drink.

To find out after how long will 80% of the target market have bought the soft drink, we need to solve the equation f(t) = 80.

Given that the function for the percentage of the target market that buys a soft drink is:

f(t) = 100(1 - e^(-0.01t))

We can substitute f(t) with 80:

80 = 100(1 - e^(-0.01t))

To solve for t, we can rearrange the equation:

1 - e^(-0.01t) = 0.8

Now, let's solve for t:

e^(-0.01t) = 0.2

To eliminate the exponent, take the natural logarithm (ln) of both sides:

ln(e^(-0.01t)) = ln(0.2)

-0.01t = ln(0.2)

Now, divide both sides by -0.01 to solve for t:

t = ln(0.2) / -0.01

Using a calculator, evaluate ln(0.2) / -0.01:

t ≈ 138.629

Therefore, after approximately 138.629 days, 80% of the target market will have bought the soft drink.

.8 = 1 - e^-.01t

-.2 = -e^-.01 t

ln .2 = -.01 t

t = -(ln .2)/.01 = 161