# 4.6: Homework- Optimization

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1. Samantha has some whiskey at a party, and (being a science and math geek) estimates her blood alcohol content (BAC) follows the function:

$$BAC(t) = \frac{0.25 t}{e^t},$$

where $$t$$ is measured in hours after her first drink. Graph this function, and determine the following using a derivative:

1. How quickly is her BAC increasing (or decreasing) 15 minutes after her first drink?
$$BAC'(t) = \frac{0.25 - 0.25t}{e^t}$$, (grams per dL per hour).
ans
2. How quickly is her BAC increasing (or decreasing) 1 hour after her first drink?
$$0$$ change
ans
3. How quickly is her BAC increasing (or decreasing) 2 hours after her first drink?
$$\approx 0.034$$ grams per dL per hour
ans
2. Graph each function over the given interval. Use calculus to determine the location of all global and local mins and maxes.
1. $$f(x) = -x^2 + 5x - 2$$ on the interval $$[0, 5]$$.
Local and Global Mins: $$(0, -2), (5, -2)$$, Local and global max: $$(2.5, 4.25)$$
ans
2. $$f(x) = x^2 - 6x + 10$$ on the interval $$[2, 6]$$.
Local min: $$(2, 2)$$, global and local min: $$(3, 1)$$, local and global max: $$(6, 10)$$
ans
3. $$f(x) = x^3 - 6x^2 + 11x - 6$$ on the interval $$[0, 3]$$.
Local and global maximum at $$(1.42, 0.38 )$$, Local min: $$(2.58,-0.38 )$$, local and global minimum: $$(0, -6)$$, local maximum: $$(3, -2)$$
ans
4. $$f(x) = x^3-5 x^2+8 x-4$$ on the interval $$[0, 2.5]$$.
Local max at $$x = 0.75$$, local min at $$x = 2$$, global max at $$x = 2.5$$, global min at $$x = 0$$.
ans
5. $$f(x) = x^4 - 16x$$ on the interval $$[0, 3]$$.
Local max at $$x = 0$$, global min at $$x = \sqrt[3]{4}$$, global max at $$x = 3$$
ans
6. $$f(x) = \frac{x^2}{x + 1}$$ on the interval $$[-3, 3]$$.
Local min at $$x = -3$$, local max at $$x = -2$$, local min at $$x = 0$$, local max at $$x = 3$$
ans
3. Using a chemotherapy drug on a petri-dish of cancer cells, it is found that $$P(x)$$ percent more of the cancer cells are killed using $$x$$ milligrams of drug per square centimeter than healthy cells, where $$x$$ ranges from $$0$$ to $$4$$. It is thought

$$P(x) = x^3-8 x^2+16 x$$

For what value of $$x$$ is $$P(x)$$ maximized?

ans
4. Bananas as we know them may be doomed! Suppose the fungus Tropical Race 4 mentioned in the article is killing off bananas on an island in Jamaica. The number of viable banana farms starts at $$16000$$, with $$800$$ being forced to close per year. But new banana farms are according to the function $$20e^{0.3t}$$ with new varieties immune to the fungus ($$t$$ measured in years). So the total number of viable banana farms is

$$V(t) = 16000- 800t + 20e^{0.3t}$$

on the interval $$[0, 20]$$. At what point is the number of banana farms minimized? What is the number of viable banana farms at this point?

This function is minimized at $$t \approx 16.3$$ with the number of banana farms at $$5619$$
ans
5. The area of a rectangle is length times width. A farmer needs to build a pig pen against the side of the barn using $$20$$ meters of fence. What is the maximum amount of area he can enclose?
6. Watch the KhanAcademy videos on maximizing the area of a box:
Optimizing Box Volume Graphically and
Optimizing Box Volume Analytically
7. An open-topped box is formed by removing the square corners of sidelength $$x$$ off of a $$40$$ in by $$80$$ in piece of cardboard, and folding each side up. What value of $$x$$ maximizes the volume of the box?
$$x \approx 8.45$$
ans
8. The height and radius of a cone together add to $$5$$ inches. What value of the radius maximizes the volume? The volume is given by $$V = \frac{1}{3} \pi r^2 h$$.
in
ans

This page titled 4.6: Homework- Optimization is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.