MAT 1730 – Exam 3
SOLUTIONS
1. Solve the equation[pic]. You will get no marks for a calculator answer.
[pic]
Checking in the ORIGINAL equation. On the left hand side:
[pic]
So, the solution to the equation is [pic]
2. Solve the equation[pic]. You will get no marks for a calculator answer.
[pic]
So, POSSIBLE solutions are [pic]. However, neither satisfies the original equation because you cannot take the logarithm of a negative number on the right hand side of the equation. Therefore, there is NO SOLUTION,
3. Solve the equation [pic]for t . You will get no marks for a calculator answer.
[pic]
This value of t satisfies the original equation.
4. The number of bacteria in a culture is increasing according to the law of exponential growth
( [pic] for some constants a, b). After 3 hours, there are 100 bacteria, and after 5 hours there are 400 bacteria. How many bacteria will there be after 6 hours?
When t = 3, P = 100. So, [pic]
When t = 5, P = 400. So, [pic]
Dividing. We get
[pic]
So, [pic]. Now substitute for b in [pic] to get [pic], which means
[pic]
When t = 6, [pic]
5. The demand, x, for a hand-held electronic organizer when the price is p (in dollars) is given by the equation[pic]. Find the demand for the product when the price is $600.
We have to solve the equation [pic] for x
[pic]
This value of x satisfies the original equation.
6. The value, V, (in millions of dollars) of a famous painting can be modeled by the equation[pic], where t represents the year, with t = 0 corresponding to 2000. In 2008, the painting was sold for $65,000,000. Find the value of k, and predict the value of the painting in 2014.
When t = 8, V = 65. So, [pic]
Solving this equation for k, [pic]
Substituting for k, [pic]
In 2014, t = 14, so [pic]and the painting would be worth...
