Difference between revisions of "009A Sample Midterm 3, Problem 2"
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!Step 1: | !Step 1: | ||
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| − | | | + | |Let <math>v(t)</math> be the velocity of the object at time <math>t.</math> |
| + | |- | ||
| + | |Then, we have | ||
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| − | | | + | | <math>\begin{array}{rcl} |
| + | \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{s(t)-s(3)}{t-3}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+200-(-4.9(9)+200)}{t-3}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}.} | ||
| + | \end{array}</math> | ||
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!Step 2: | !Step 2: | ||
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| − | | | + | |Now, we factor the numerator to get |
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| + | <math>\begin{array}{rcl} | ||
| + | \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t^2-9)}{t-3}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t-3)(t+3)}{(t-3)}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{t\rightarrow 3} -4.9(t+3)}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{6(-4.9) \text{ meters/second}.} | ||
| + | \end{array}</math> | ||
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!Final Answer: | !Final Answer: | ||
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| − | |'''(a)''' | + | | '''(a)''' <math>6(-4.9) \text{ meters/second}</math> |
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|'''(b)''' | |'''(b)''' | ||
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[[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 14:10, 17 February 2017
The position function gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time seconds is given by:
- a) Find the velocity of the object when 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 t=3}
- b) At what velocity will the object impact the ground?
| Foundations: |
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Solution:
(a)
| Step 1: |
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| Let 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 v(t)} be the velocity of the object at time 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 t.} |
| Then, we have |
| 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 \begin{array}{rcl} \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{s(t)-s(3)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+200-(-4.9(9)+200)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}.} \end{array}} |
| Step 2: |
|---|
| Now, we factor the numerator to get |
|
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 \begin{array}{rcl} \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t^2-9)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t-3)(t+3)}{(t-3)}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} -4.9(t+3)}\\ &&\\ & = & \displaystyle{6(-4.9) \text{ meters/second}.} \end{array}} |
(b)
| Step 1: |
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| Step 2: |
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| Final Answer: |
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| (a) 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 6(-4.9) \text{ meters/second}} |
| (b) |