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Question 1 A spring that has a natural/resting length of 5 cm is stretched to a length of 25 cm by a 5 N force. How much work was required to do that?
Step 1: Stretch distance is \(25-5=20\) cm \(=0.2\) m. Step 2: Use Hooke's Law \(F=kx\). Step 3: \(k=5/0.2=25\). Step 4: \(W=\frac12kx^2=\frac12(25)(0.2)^2=0.5\) J. Answer: A. 0.5 J
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Question 2 The average value of \(f(x)=\csc x\) on \([\frac{\pi}{4},\frac{\pi}{2}]\) is \(1.12\).
Step 1: Use \(f_{avg}=\frac1{b-a}\int_a^bf(x)dx\). Step 2: Compute \(\frac{4}{\pi}\int_{\pi/4}^{\pi/2}\csc x\,dx\). Step 3: Use \(\int\csc xdx=\ln|\tan(x/2)|+C\). Step 4: Evaluate to obtain approximately \(1.12\). Step 5: Statement is true. Answer: B. True
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Question 3 If a spring is oscillating according to \(v(t)=\cos t\) (ft/s), then its displacement over its first \(2\pi\) seconds is:
Step 1: Displacement is \(\int_0^{2\pi}\cos t\,dt\). Step 2: Evaluate \([\sin t]_0^{2\pi}=0\). Step 3: Therefore displacement is 0 ft. Answer: B. 0 ft
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Question 4 The distance a particle travels in its first 4 seconds if \(v(t)=-t^2+4\) (ft/s) is:
Step 1: Distance is \(\int_0^4|v(t)|dt\). Step 2: Velocity changes sign at \(t=2\). Step 3: Compute \(\int_0^2(-t^2+4)dt-\int_2^4(-t^2+4)dt\). Step 4: The values are \(\frac{16}{3}\) and \(-\frac{32}{3}\). Step 5: Total distance is \(16\) ft. Answer: B. 16 ft
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