Difference between revisions of "009A Sample Final 2, Problem 9"
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| <math>\frac{2500 \text{ miles}}{5.5 \text{ hrs}}\approx 454.5 \text{ miles/hr}.</math> | | <math>\frac{2500 \text{ miles}}{5.5 \text{ hrs}}\approx 454.5 \text{ miles/hr}.</math> | ||
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| − | | | + | |In order to average this speed, the plane had to go from 0mph, up to full speed, past 454.5mph, and then it had to go back down to 0mph to land. |
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| − | | | + | |This means that there will be at least two times where the plane of the speed is 400mph. |
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!Final Answer: | !Final Answer: | ||
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| − | | | + | |See solution above. |
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[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 19:35, 7 March 2017
A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using the mean value theorem to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.
| Foundations: |
|---|
| Mean Value Theorem |
| Suppose is a function that satisfies the following: |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous on the closed interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b].} |
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is differentiable on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b).} |
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Then, there is a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<c<b} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}.} |
Solution:
| Step 1: |
|---|
| On average the plane flew |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2500 \text{ miles}}{5.5 \text{ hrs}}\approx 454.5 \text{ miles/hr}.} |
| Step 2: |
|---|
| In order to average this speed, the plane had to go from 0mph, up to full speed, past 454.5mph, and then it had to go back down to 0mph to land. |
| This means that there will be at least two times where the plane of the speed is 400mph. |
| Final Answer: |
|---|
| See solution above. |