Tutor profile: Jessie D.
Subject: Linear Programming
Pete the Plumber charges $230 for a job that takes 4 hours and $335 for a job that takes 7 hours. State the equation relating the amount he charges (C) and his hours worked (t).
To start with, it's useful to remember that linear equations take the form y = mx+c. We need to find values for "m" and "c" and may need to replace our variables "y" and "x" with pronumerals suited to the situation. Let's consider the information we have been given... When Pete works 4 hours, he charges $230. When he works 7 hours, he charges $335, We have been given 2 points! (4, 230) and (7, 335). We need to start by finding the gradient, using... m = rise/run = (y2-y1)/(x2-x1) Label our points (x1, y1) --> (4, 230) & (x2, y2) --> (7, 335) Substitute and simplify/calculate: m = (335-230)/(7-4) = 105/3 = 35 Our gradient is 35 - this also means he charges $35 per hour Now to find our y-intercept, "c". Substitute either of our points (let's use (x1, y1)) and m into y=mx+c m = 35 and (4, 230) --> 230 = 35*4 + c 230 = 140 + c 230 - 140 = c c = 90 Our y-intercept is 90 - this means that he charges $90 as the call-out fee (even if he works 0 hours!) Substitute m & c into y=mx+c y = 35x + 90 Finally - re-write with suitable pronumerals C = 35t + 90
From the top of a building, 202 m high, the angle of depression to another building, 120 m away, is 25 degrees. How high is this second building?
For application problems, it is always best to start by drawing a diagram (which I will assist my students with). Now, remember when labelling your angle of depression - it is from the point looking DOWN. But angles of depression are always made with the horizontal - so you will need to draw a dotted line to show this angle. The diagram you have drawn should be a trapezium - we need to find a right-angled triangle in this. Let's start from the top of the smallest building to the top of the highest. We know that the distance between the buildings is 120m - we can label the horizontal line of our triangle with this. We also have the angle of depression which sits on the outside of our triangle - our knowledge of alternate "Z" angles means that the angle at the bottom of the triangle is also 25 degrees. What are we trying to find? The height. So let's label our vertical side with "x". Now to solve: 1. Label our sides - O, A & H, in relation to our 25 degrees. We have A = 120, and we want O = x 2. Choose the appropriate trigonometric ratio - SOH CAH TOA A & O - we need to use TOA 3. Write out the trigonometric ratio and substitute. 4. Solve for "x" x = 55.96m Re-read the question: "How high is this second building? " This answer has only given us how much TALLER the second building is - we need to add our answer to the smaller building's height. Height = 202 + 55.96 = 257.96m *Don't forget the units in your answer! Note: I will be able to provide more guidance through diagrams and correct use of mathematical notation during sessions
The sum of five consecutive odd numbers is 585. What are the numbers?
Start by letting our first number equal to a pronumeral, let's say "x". This means that our second number will be equal to "x+2" (as it is 2 more than our first number, being the next ODD number). Our third number will be equal to "x+4", and so on. This gives us... x, x+2, x+4, x+6, x+8 - these are our 5 numbers. Remember, "sum" means "plus" or "addition"... so we have: x + x+2 + x+4 + x+6 + x+8 = 585 We can simplify this by collecting like terms: 5x + 20 = 585 Now we have a basic linear equation to solve - our goal is to isolate "x". Step 1: Take 20 from both sides 5x + 20 (-20) = 585 - 20 5x = 565 Step 2: Divide both sides by 5 5x (/5) = 565/5 x = 113 Always go back and check your answer: 113 + 115 + 117 + 119 + 121 = 585 which is correct. Re-read the question... "What are the numbers?" Answer: 113, 115, 117, 119, 121
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