Question 1: y= 2( ) +5 y= The line intersect the circle at points (-1.07, 2.85) and (-3.73,-2.45)

Transcription

1 Question 1: At what points (x,y) does the line with equation y = x+5 intersect the circle with radius 3 and center (-, 0)? The equation of the circle is (x + ) + y = 3 We can then substitute the y for x+5 (x - (-)) + ( x + 5) - 9 = 0 5 x + 4 x + 0 = 0 After using the quadratic formula, to get the x values and We plug that into y=x+5 y= ( ) +5 y=.8533 y= ( ) +5 y= -.45 The line intersect the circle at points (-1.07,.85) and (-3.73,-.45) I don t know why there is so much space in between the two questions.

2 111 FinalMTH150.nb Question : (1) Convert the angle 30 to radians. I am going to use the degrees to radians formula, which is Degrees * 30 * π = π π 180 = Radians 11 π () Convert the angle from radians to degrees 6 I am going to use the radians to degrees formula, which is Radians * 11 π 6 11 π 6 * * 180 = 1980 π π = π = Degrees Question 3: (1) If cos(θ) = 9 and 0 < θ < π find the value of sin(θ). In order to solve this problem we have to use the Pythagorean theorem, which is a + b = c + b = b = 81 b = 77 Now that we know the opposite side, we plug in the rest of the numbers. sin(θ) = 77 sin(θ) = () If sin(θ) = and π < θ < 3 π find the value of cos(θ). In order to solve this problem we have to use the Pythagorean theorem, which is a + b = c -1 + b = b = 16 b = 15 Now that we know the adjacent side, we plug in the rest of the numbers. cos(θ) = cos(θ) = Question 4: A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a certain amount of money per megabyte (MB) of data used on

3 FinalMTH150.nb 1113 the phone: If a customer uses 0 MB, the monthly cost will be $11.0 If the customer uses 130 MB, the monthly cost will be $17.80 (1) Find a linear equation for the monthly cost of the data plan as a function of the number of MB used. I am going to make the equation in the form of y=mx+b, monthly cost = (amount of money * MB) + monthly fee monthly cost = (0.06 * x) + 10 () Interpret the slope and vertical intercept of the equation. The slope is 0.06 and the vertical intercept is 10. (3) Use your equation to find the total monthly cost if 50 MB are used. monthly cost = (0.06 * 50) + 10 monthly cost = $5 Question 5: A business purchases $15,000 of office furniture which depreciates at a constant rate of 1% each year. Find the residual value of the furniture 6 years after purchase. A = P(1 + r /n) nt A = 15,000(1-0.1) 6 A = 58, Question 6: A radioactive substance decays exponentially. A scientist begins with 110 milligrams of the radioactive substance. After 31 hours, 55 mg of the substance remains. How many milligrams will remain after 4 hours? In order to solve this problem, we have to find the half-life and then plug in the time and the initial amount. 110 * (-4/31) 55 11/31 N 55 11/

4 4 111 FinalMTH150.nb Question 7: (1) Write a formula for the function g(x) that results when the function f(x) = x +1 is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units. g(x) = 6 x + 4) - () Plot f(x) and g(x) on the same plot. f[x_] := x + 1 g[x_] := 3 (x + 4) - Plot[{f[x], g[x]}, {x, -10, 10}, PlotStyle {Red, Green}] Question 8: A scientist begins with 100 mg of a radioactive substance. After 4 hours, it has decayed to 80 mg. How long after the process began will it take to decay to 15 mg? In order to solve this problem, we have to find the half-life using this formula A = P -t/h 100-4/h = 80-4/h = 80/100-4/h = 4/5 log( -4/h ) = log(4/5) -4/h = log() log(4/5) h = -4 log() log(4/5) h = 1.45

5 FinalMTH150.nb * (-15/1.45) Question 9: In the diagram below angles of 30 and -30 are shown, together with cos(30 ) - in blue - and sin(-30 ) - in red: (1) Given that cos(30 ) = and sin(-30 ) = - 1 use the addition formula for the sin of a sum of two angles, and the fact that 15 = to calculate sin(15 ) Sin(45-30) = Sin(45)Cos(-30) + Cos(45)Sin(-30) () Check you answer with Mathematica Sin[45 Degree] * Cos[-30 Degree] + Cos[45 Degree] * Sin[-30 Degree] // N Question 10: (1) Plot y = sin( π x 4 ) and y = 1 on the same plot, and so determine for how many values of x between -π and π we have sin( π x 4 ) = 1

6 6 111 FinalMTH150.nb Plot Sin[(π * x) / 4], 1, {x, - π, π} () Find the exact value of x between 0 and for which sin( π x 4 ) = 1 x = /3 Question 11: The more you study for a certain exam, the better your performance on it: If you study for 10 hours, your score will be 65%. If you study for 0 hours, your score will be 95%. You can get as close as you want to a perfect score just by studying long enough. Assume your percentage score, p(n), is a function of the number of hours, n, that you study in the form n+b p(n) = n+d (1) If you want a score of 80%, how long do you need to study? If 0 = 95 and 10 = 65, then 15 must = hours () Plot a graph showing the behavior of p(n) as a function of n for 0 n 40. Plot n n, {n, 0, 40}

7 FinalMTH150.nb 1117 Question 1: The number of people P(t) in a town who have heard a rumor after t days can be modeled by the 500 function P(t) = 1+49 e -0.7 t (1) Plot a graph of this equation. Plot E^ 0.7 * t, {t, -, 4} () How many people started the rumor? 10 people (3) How many people have heard the rumor after 3 days? 70 people (4) How long will it take until 300 people have heard the rumor? 6. days