Question

The rate of consumption of liquid chocolate in Switzerland is given by R(t)=ceᵏᵗ , where R is measured in millions of gallons per year and t is measured in years from the beginning of 1990. The consumption rate doubles every 9 years and the consumption rate at the beginning of 1990 was 2 million gallons per year. Find c and k.

Answer 1

The constant c equals 2 million gallons per year since the initial consumption was 2 million gallons. The constant k is found to be approximately 0.077 by using the fact that the consumption rate doubles every 9 years.

Step-by-step explanation:

To find the values of c and k for the function R(t)=cekt, where R is the rate of consumption of liquid chocolate in Switzerland, and t is measured in years from 1990, we use the given information.

First, we know that the consumption rate at the beginning of 1990 (t=0) was 2 million gallons per year, so we can set up the equation R(0)=c to solve for c. This gives us:

R(0) = c·ek·0 = c·1

R(0) = 2

Therefore, c = 2 million gallons per year.

Next, the consumption rate doubles every 9 years, which means R(9) = 2R(0) or 2c = cek·9. We use this information to find k, dividing both sides by c and taking the natural logarithm:

2 = ek·9

ln(2) = ln(ek·9)

ln(2) = 9k

k = ln(2)/9

Substituting ln(2) which is approximately 0.693, gives us the value for k as:

k ≈ 0.693/9 ≈ 0.077

In conclusion, we have determined the values c = 2 and k ≈ 0.077 for the given function of chocolate consumption in Switzerland.