Further problems 1

Exercise 1
  1. You have inherited land that was purchased for 3000030 000 dollars in 1960. The value of the land increased by approximately 5%5\% per year. What is the approximate value of the land in the year 20112011?
  2. During normal breathing, about 12%12\% of the air in the lungs is replaced after one breath. Write an exponential decay model for the amount of the original air left in the lungs if the initial amount of air in the lungs is 500 mL. How much of the original air is present after 24 breaths?
  3. An adult takes 400 mg of Ibuprofen. After every two hours, the amount of Ibuprofen in the system decreases by about 29%29\%. How much Ibuprofen is left after 15 minutes?
  4. The number of germs grow exponentially. At 8:00 the number was 2400, at 12:00 it was 36000. How many germs are there at 9:00?
  5. Determine the function equation of exponential functions shown below:
  6. Suppose you deposited 700700 dollars into an account that pays 5.80%5.80\% interest per annum. How much money will you have in the account at the end of 5 years if the interest is compounded quarterly, that is, each quarter of a year you get 5.80/4=1.45%5.80/4=1.45\% interest?
  7. Moore's famous law states that (roughly speaking), every 1.5 years computer power (how fast a computer can process information) doubles. By how much does computer power increase every 3 years, and every half a year?
  8. Energy consumption doubles every twenty years. By how many percent does energy consumption increase every year?
  9. The weight of a dog in week two is 0.72kg0.72kg, in week eight it is 2.18kg2.18kg. Determine the weight of the dog in week 2121
    1. if growth is linear. What is the slope of the linear function describing the growth?
    2. if growth is exponential. Also, what is the growth factor and percent increase for every week?
  10. Consider the points A(−2∣5)A(-2\vert 5) and B(2.5∣1.7)B(2.5\vert 1.7).
    1. Determine the function equation of the straight line ff that passes through AA and BB. Is the point P(100∣−70)P(100\vert -70) on the straight line?
    2. Determine the function equation of the exponential function ff whose graph passes through AA and BB. Is the point P(10∣0.084)P(10\vert 0.084) on the graph?
  11. The half-life of the radioactive substance radon-222 is 3.83.8 days. This means that every 3.8 days the mass of the substance is halved. By how many percent reduces the substance every

    1. day?

    2. minute?

    3. week?

Solution
  1. should be 9.
    1. f(x)=0.72+1.46⋅x−26=0.243‾x+0.23‾f(x)=0.72+1.46\cdot \frac{x-2}{6} = 0.24\overline{3}x+0.2\overline{3}, thus the slope is a=0.243‾‾a=\underline{0.24\overline{3}} and the weight at x=21x=21 is f(21)=5.343kg‾f(21)=\underline{5.343kg}
    2. f(x)=0.72⋅3.027‾x−26f(x)=0.72\cdot 3.02\overline{7}^\frac{x-2}{6}, thus the weight at x=21x=21 is f(21)=24.037kg‾f(21)=\underline{24.037kg}. Because f(3)=0.866kgf(3)=0.866 kg, the growth factor for every week is u=0.8660.72=1.2027‾u=\frac{0.866}{0.72}=\underline{1.2027}, which corresponds to a percent increase of 20.27%‾\underline{20.27\%}.
  2. should be 10.
    1. f(x)=−0.73‾x+3.53‾f(x)=-0.7\overline{3}x+3.5\overline{3}. Because of f(100)=−69.8f(100)=-69.8, point P(100∣−70)P(100\vert -70) is not on the straight line.
    2. f(x)=5⋅0.34x+24.5f(x)=5\cdot 0.34^{\frac{x+2}{4.5}}. Because of f(10)=0.2185...f(10)=0.2185..., the point P(10∣0.084)P(10\vert 0.084) is not on the graph of ff.
  3. should be 11.
    1. growth factor for every day is u=0.513.8=0.83326u=0.5^{\frac{1}{3.8}}=0.83326, this corresponds to p=16.673%‾p=\underline{16.673\%}.
    2. growth factor for every minute is u=0.513.8⋅24⋅60=0.99987u=0.5^{\frac{1}{3.8\cdot 24\cdot 60}}=0.99987. This corresponds to p=0.0126%‾p=\underline{0.0126\%}
    3. growth factor for every week is u=0.573.8=0.27891u=0.5^\frac{7}{3.8}=0.27891, which corresponds to p=72.108%‾p=\underline{72.108\%}.