Quadratic Applications with Solutions

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STEPS to Solve Word Problems Read and reread the problem. Try to picture the problem. Ask what you are looking for? Determine which formula you should use to solve the problem. 5. Use the formula to solve and conclude with an answer in sentence form.

Page 34 # 26 What are we trying to find out? A submarine travelling in a parabolic arc ascends to the surface. The path of the submarine is described by where x represents the time in minutes and y represents the depth of the submarine in metres. a) How deep is the submarine initially? What are we trying to find out? The depth of the submarine at time 0. So we have to substitute 0 for x in the function. Initially, the sub is 50 m below the surface.

Page 34 # 26 What are we trying to find out? A submarine travelling in a parabolic arc ascends to the surface. The path of the submarine is described by where x represents the time in minutes and y represents the depth of the submarine in metres. b) For how long is the submarine underwater? What are we trying to find out? The positive root of the quadratic equation which will tell where the sub surfaces on the x “time” line. The submarine is underwater for approximately 8.1 minutes.

Page 34 # 26 What are we trying to find out? A submarine travelling in a parabolic arc ascends to the surface. The path of the submarine is described by where x represents the time in minutes and y represents the depth of the submarine in metres. c) What is the deepest position from the surface? What are we trying to find out? The minimum value of the function that occurs at the vertex. Use the Vertex Formula Solve for y The sub’s deepest position was 62.5 m below the surface.

Page 44 # 6 What are we trying to find out? The manager of a peach orchard is trying to decide when to arrange for picking the peaches.If they are picked now, the average yield per tree will be 100 kg, which can be sold for 40¢ per kg. Past experience shows that the yield per tree will increase about 5 kg weekly, while the price will decrease by 1¢ per kg weekly. a) When should the peaches be picked in order to maximize the revenue, and what will the maximum revenue be? What are we trying to find out? We are looking for the maximum revenue so we need to determine the vertex, but first we must determine the quadratic function for Revenue. Let x represent the number of weeks before picking. Revenue = Price X Quantity Our quadratic function is

Page 44 # 6 What are we trying to find out? The manager of a peach orchard is trying to decide when to arrange for picking the peaches.If they are picked now, the average yield per tree will be 100 kg, which can be sold for 40¢ per kg. Past experience shows that the yield per tree will increase about 5 kg weekly, while the price will decrease by 1¢ per kg weekly. a) When should the peaches be picked in order to maximize the revenue, and what will the maximum revenue be? What are we trying to find out? We are looking for the maximum revenue so we need to determine the vertex, but first we must determine the quadratic function for Revenue. Use the Vertex Formula Solve for y The peaches should be picked in 5 weeks for a maximum revenue of $45.

Page 44 # 6 What are we trying to find out? The manager of a peach orchard is trying to decide when to arrange for picking the peaches.If they are picked now, the average yield per tree will be 100 kg, which can be sold for 40¢ per kg. Past experience shows that the yield per tree will increase about 5 kg weekly, while the price will decrease by 1¢ per kg weekly. b) When will the revenue be zero? What are we trying to find out? We are looking for the root of the quadratic equation. Let’s use the Quadratic Formula a=-.05 b=1 c=40 inadmissable If the peaches are picked in 40 weeks, the revenue will be $0.

Page 54 # 48 What are we trying to find out? A chemical power plant is rectangular and has a length of 100 m and a width of 60 m. A safety zone of uniform width surrounds the plant. If the area of the safety zone equals the area of the plant, what is the width of the safety zone? What are we trying to find out? We are finding the width of the safety zone, which we will represent by x. So we will finding the roots of the equation we determine. 60 m 100 m Area of Chemical Plant = 60 X 100 = 6000 m2 Let x represent the width of the safety zone Area of Safety Zone = Area of Total Space – Area of Chemical Plant = (100 + 2x)(60 + 2x) – (100)(60) = (6000 + 320x + 4x2 ) – (6000) = 320x + 4x2

Page 54 # 48 A chemical power plant is rectangular and has a length of 100 m and a width of 60 m. A safety zone of uniform width surrounds the plant. If the area of the safety zone equals the area of the plant, what is the width of the safety zone? So, Area of the Safety Zone = 320x + 4x2 But the Area of the Safety Zone must equal the Area of the Chemical Plant 60 m 100 m Solve for x. This solution is inadmissible. The width of the safety zone is approximately 15.68 m.

Look for clues. Remember to look for clues in the Word Problem. Which Critical Point will give you the Answer?

When you’ve got the Quadratic Function, determine which formula to use and WORK IT OUT!

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