MA105 Grantham University Week 7 Logarithm Problems Paper

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oqe513

Mathematics

MA105

Grantham University

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I just need help with the answers. I did not have a group for the 2 questions that said it was a group project.

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· Problem 6 o What are the coordinates of the point (x,y)? o Give the equation for the path of the ball, showing all work. Explain why you do each step. o Include a screenshot of the graph of your equation (use Desmos.com), verifying that it goes through the given points. · Problem 10 o The total time T (in hours) of the trip as a function of the distance x (in miles). Make sure you show all your work to come up with this equation, and explain each step and what it represents. o The domain of the function. Explain why this domain makes sense, and show work as needed. o Graph the function using the graphing utility on Desmos.com. Take a screenshot and make sure it is included in your document. The graph should be focused over the domain of the function. o Find the value of x that minimizes T. Desmos.com will note this ordered pair for you automatically. If it does not, then you can click various points of the graph and it will display the ordered pair. o What does the ordered pair of the minimum for T represent in our situation? I.E. what does the value of x tell us, and what does the value of T tell us? Write a brief paragraph interpreting these values. · Reflection on the project as a whole o Process for each problem § Did your idea for solving this work? § Did you have to get help or try different methods? § Was this problem easy? Hard? § Share anything else you have to say about the problems o Working with a partner § Were you and your partner able to communicate well? § What technology did you use to meet and communicate? § How did you split up work? Did that work well? § Share anything else you have to say o Your successes and failures throughout this entire project Please watch the weekly discussion video (https://www.youtube.com/watch?v=IUetWQ_9BT0&feature=youtu.be). This week you were introduced to the Euler's number e, a mathematical constant. Another well-known constant is pi. Do some research on other important constants in mathematics. A good constant has a specific name and is important in mathematical formulas. Do not choose pi or e. Share with us what you have found out about the number; for example, its history, some trivial facts about the number, and/or how it is applied. Be sure to include your sources so others can do further research. To get you started in your research, below are some sources you can use: • http://www.popularmechanics.com/flight/g163/13-most-important-numbers-in-the-universe/ • https://en.wikipedia.org/wiki/Mathematical_constant#Table_of_selected_mathematical_consta nts
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Explanation & Answer

The solutions are ready, please ask if something is unclear. Pdf file is the same as Word just in case.

1. 𝟒𝒙 (positive and increasing everywhere) (A).

2. By definition of logarithm this means 𝟓𝟐 = 𝟐𝟓.

𝑥3

3. log 8 𝑦 3 𝑧 3 = 𝟑 𝐥𝐨𝐠 𝟖 𝒙 − 𝟑 𝐥𝐨𝐠 𝟖 𝒚 − 𝟑 𝐥𝐨𝐠 𝟖 𝒛 (D).

4. The exponents must be the same, i.e.
−4𝑥 2 = 3𝑥 2 − 14𝑥, 7𝑥 2 − 14𝑥 = 7𝑥(𝑥 − 2) = 0,
so 𝑥 = 𝟎 or 𝑥 = 𝟐.

5. The last,
The graphs are similar, they differ only by vertical position. Note that 𝑦(−1) = 5 + log 1 = 5
and 𝑦(0) = 5 + log 2 ∈ (5, 6).

6. True. When 𝑥 → −∞, 𝑦(𝑥) → −5, so 𝑦 = −5 is a (horizontal) asymptote.

7. Arguments of logs must be the same, i.e. 1 − 𝑥 = 100, so 𝑥 = 1 − 100 = −𝟗𝟗 (D).

𝑥

𝒙 𝟑

8. 3(log 𝑥 − log 𝑦) = 3 log 𝑦 = 𝐥𝐨𝐠 (𝒚) (E).

9. 𝑒 0 = 1 so the factor before exponent must be 5. Because 𝑦(4) = 1 < 5, the exponent is
decreasing. So, the answer is 𝑦 = 𝟓𝒆−𝟎.𝟒𝟎𝟐𝟒𝒙 (A).

ln 8

10. Apply ln to both sides and obtain 𝑥 ln 5 = ln 8, i.e. 𝑥 = ln 5 ≈ 𝟏. 𝟐𝟗𝟐 (E).

11. The function is always decreasing and tends to minus infinity, so B

12. By definition of logarithm this means 𝐥𝐨𝐠 𝟐 𝟖 = 𝟑.

13. It is 𝑃(6) = 1200𝑒 0.052∙6 = 1200𝑒 0.312 ≈ 1200 ∙ 1.366 ≈ 1639 (round to the closest
integer because it is the number of bacteria).

14. Apply exponent to both sides and obtain 2𝑥 = 𝑒 3 , 𝑥 =

𝑒3
2

≈ 10.043.

15. False. Consider 𝑥 = 8, the equation would be ln 1 = ln 8 − ln 7. But it is incorrect
because the left part is zero while the right part is not.

6. Miniature Golf.
a. The point of reflection has the y-coordinate of 𝟖 (𝑓𝑡) as the entire upper side h...


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