Question 1 Solutions
- 2. T Point “ T ” is where the telephone is located. We know that the diameter of the circle is 3.4 meters because the problem mentioned that the father’s farthest distance from the phone is 3.4 meters. If the diameter of the circle is 3.4 meters, therefore the radius is 1.7 meters. 1.7m
- 3. T 1.7m In order to solve for the circular distance between the father and the phone when it rang, we need to know how long the it would take for the father to make one complete revolution… Period: 2 mins. 60 secs 120 secs 3.1414 seconds 38.2 rev 1 min 38.2 rev revolutions x x =
- 4. Now that we have the period, we can now make our function, either sine or cosine using the “DABC” pneumonic … SINE FUNCTION A – 1.7 (it is the amplitude because it is half of 3.4m, which is the max) B – 2π 2 (B is equal to 2π / Period) 3.1414 C – 3.1414/4 = 0.78535 (phase shift of 0.78535 to the right) D – 1.7 (The Sinusoidal axis) Therefore… d (t) = 1.7sin2(t – 0.78535) + 1.7
- 5. OR COSINE FUNCTION A – -1.7 (the graph is rotated along the x-axis because the minimum is at the origin) B – 2 (the period is the same) C – none (there is no phase shift because the minimum is at the origin) D – 1.7 (The Sinusoidal axis) Therefore… d (t) = -1.7cos2(t) + 1.7
- 6. Now we have our formula(s), we can now solve for the time when the phone rang and he was 2.4 meters away from the phone. By graphing the function(s), we can zero-in on the value of the time/angle.
- 7. d (t) = 1.7sin2(t – 0.78535) + 1.7 2.4 = 1.7sin2(t – 0.78535) +1.7 0.7 = 1.7sin2(t – 0.78535) Let θ = 2(t – 0.78535) 0.7 = 1.7sin(θ) sin(θ) = 0.7/1.7 sin(θ) = 0.4118 θ = 0.4244 2(t – 0.78535) = 0.4244 2t – 1.5705 = 0.4244 2t = 1.9951 t = 0.9976 seconds - Sine Formula - substituting the value 2.4 for d(t) subtracting 1.7 from both sides substituting θ for the (t – 0.78535) substituting θ dividing both sides by 1.7 - using arcsine - re substituting (t – 0.78535) distributing 2 to (t – 0.78535) adding 1.5705 from both sides - dividing 2 from both sides
- 8. d (t) = -1.7 cos2(t) + 1.7 2.4 = -1.7 cos2(t) + 1.7 0.7 = -1.7 cos2(t) Let θ = 2t 0.7 = -1.7 cos(θ) cos(θ) = - 0.4118 θ = 1.9952 2t = 1.9952 t = 0.9976 seconds - Cosine Formula - substituting the value 2.4 for d(t) - subtracting 1.7 from both sides - substituting θ for the 2(t) - substituting θ - dividing both sides by 1.7 - using arccosine - re substituting 2(t) - dividing 2 from both sides If we use the cosine formula…
- 9. Now that we have the time when the father was 2.4 meters away from the telephone, we can use this to find the circular distance/arc length between the father and the phone when it rang…
- 10. Arc Length Between the father and the phone Radian angle (time) Arc Length (L) 2π 2π(radius) 0.9976 L 2π 2π (1.7) 3.3918π L 2π 1.6959 meters L = = = =