Chapter 3 – Solving Quadratics Word Problems Where You Have to Create the Equation

Agenda Movement problems (creating the equation)* Engineering problems (creating the equation)* Revenue problems (creating the equation)* Area problems (creating the equation)* Integer problems (creating the equation) Triangle problems (creating the equation) * Indicates done within this lecture

Mental Health Break

Quadratic Word Problems Movement Creating the Equation – Ex. P Angus is playing golf. The diagram (not to scale, shows him making a perfect shot to the hole. Determine the height of the ball when it is 15m from the hole by using the information in the diagram to determine a quadratic relation for height v. distance travelled

Quadratic Word Problems Movement Creating the Equation – Ex. P

Quadratic Word Problems Movement Creating the Equation – Ex. P

Quadratic Word Problems Movement Creating the Equation – Ex. P

Quadratic Word Problems Movement Creating the Equation – Ex. P To find the height we will need the equation To find the equation we need the zeros and one other point Zeros… x of vertex… y of vertex…

Quadratic Word Problems Movement Creating the Equation – Ex. P To find the height we will need the equation To find the equation we need the zeros and one other point Zeros are 0 and 100 x of vertex = (0+100)/2 = 50 y of vertex… 10 (height of the tree in the middle)

Quadratic Word Problems Movement Creating the Equation – Ex. P Use y = a(x – S)(x – t) put in the zeros of 0 and 100 for S & t* put in the point (50, 10) for x & y *Note – you would end up with the same “a” if you used -50 and 50 for the zeros and made x = 0 the centre of the graph

Quadratic Word Problems Movement Creating the Equation – Ex. P y = a(x – S)(x – t) 10 = a(50 – 0)(50 – 100) 10 = a(50)(-50) 10 = a(-2500) 10/-2500 = a(-2500)/-2500 a = -0.004 y = -0.004x(x – 100) y = -0.004x²+0.4x

Quadratic Word Problems Movement Creating the Equation – Ex. P y = -0.004x²+0.4x We shouldn’t cannot plug in an x of 15m because we are 15m from where the ball lands If we did, we could argue symmetry however Ideally, we plug in 85m for x because 100m – 15m = 85m and we can use that distance to then calculate the height of the ball at that time

Quadratic Word Problems Movement Creating the Equation – Ex. P y = -0.004x² + 0.4x If we plug in x of 85m, we can calculate y (the height of the ball at that time) y = -0.004(85)² + 0.4(85) y = -0.004(7225) + 34 y = -28.9 + 34 y = 5.1m So, the height of the ball 15m from the hole is 5.1m

Quadratic Word Problems Movement Creating the Equation – Ex. P

Quadratic Word Problems Engineering Creating the Equation – Ex. P The second span of the Bluewater Bridge, in Sarnia, Ontario, is supported by a pair of steel parabolic arches. The arches are set in concrete foundations that are on opposite sides of the St. Clair River 281m apart. The top of each arch rises 71m above the river. Determine the algebraic expression that models the arch and find the height of the span 10 metres from the left side of the river.

Quadratic Word Problems Engineering Creating the Equation – Ex. P

Quadratic Word Problems Engineering Creating the Equation – Ex. P

Quadratic Word Problems Engineering Creating the Equation – Ex. P To find the height we will need the equation To find the equation we need the zeros and one other point Zeros… x of vertex… y of vertex…

Quadratic Word Problems Engineering Creating the Equation – Ex. P To find the height we will need the equation To find the equation we need the zeros and one other point Zeros are 0 and 281 x of vertex = (0+281)/2 = 140.5 y of vertex… 71 (height of the span of bridge in the middle)

Quadratic Word Problems Engineering Creating the Equation – Ex. P Use y = a(x – S)(x – t) put in the zeros of 0 and 281 for S & t* put in the point (140.5, 71) for x & y *Note – you would end up with the same “a” if you used –140.5 and 140.5 for the zeros and made x = 0 the centre of the graph

Quadratic Word Problems Engineering Creating the Equation – Ex. P y = a(x – S)(x – t) 71 = a(140.5 – 0)(140.5 – 281) 71 = a(140.5)(-140.5) 71 = a(-19740.25) 71/-19740.25 = a(71)/-19740.25 a = -0.004 y = -0.004(x - 0)(x – 281) y = -0.004x(x – 281) y = -0.004x² + 1.124x

Quadratic Word Problems Engineering Creating the Equation – Ex. P y = -0.004x²+1.124x We can plug in an x of 10m

Quadratic Word Problems Engineering Creating the Equation – Ex. P y = -0.004x² + 1.124x If we plug in x of 10m, we can calculate y (the height of the ball at that time) y = -0.004(10)² + 1.124(10) y = -0.004(100) + 11.24 y = -0.4 + 11.24 y = 10.84 So, the height of the bridge span 10m from the left side of the river is 10.84m

Quadratic Word Problems Revenue Creating the Equation Recall, R = price x # sold Profit is the revenue of a company after expenses are subtracted Breakeven takes place when profit (not revenue) is zero Maximum profit or revenue takes place at the vertex of the graph

Quadratic Word Problems Revenue Creating the Equation Icky Bum Ltd. research shows that a $0.50 increase in the price of a box of baby wipes results in 5 fewer boxes of baby wipes being sold. The usual price of $20 for a box of baby wipes results in a sale of 280 boxes What should the Icky Bum do? Should they “stick” with the current price?

Quadratic Word Problems Revenue Creating the Equation Equation set-up is always the same for this type of Revenue problem

Quadratic Word Problems Revenue Creating the Equation

Quadratic Word Problems Revenue Creating the Equation R = (orig. price + ∆ in price) (orig. # sold + ∆ in number sold) R = (20 + 0.50x)(280 – 5x)

Quadratic Word Problems Revenue Creating the Equation R = (orig. price + ∆ in price) (orig. # sold + ∆ in number sold) R = (20 + 0.50x)(280 – 5x) Our strategy is going to be to set-up the equation, find the zeros, take the average of the zeros to find the x of the vertex, and substitute x back in and solve for the vertex R which will give us the maximum revenue

Quadratic Word Problems Revenue Creating the Equation R = (20 + 0.50x)(280 – 5x) 0 = (20 + 0.50x)(280 – 5x) 20 + 0.50x = 0 and 280 – 5x = 0 (set each bracket = 0) 0.50x = -20 and -5x = -280 x = -40 and x = 56

Quadratic Word Problems Revenue Creating the Equation R = (20 + 0.50x)(280 – 5x) Zeros are -40 and 56 xv = xzero 1 + xzero 2 --------------------------- 2 xv = (-40 + 56) --------................ xv = 8

Quadratic Word Problems Revenue Creating the Equation R = (20 + 0.50x)(280 – 5x) xv = 8 R = (20 + 0.50(8))(280 – 5(8)) R = (24)(240) So, maximum revenue is $5,760 (note – previously, it was $20 x 280 = $5,600)

Quadratic Word Problems Revenue Creating the Equation

Quadratic Word Problems Area Creating the Equation When creating quadratic area equations, you are quite often working with either the area equation alone, or with the perimeter equation as well. When working with maximum area created by enclosure of so much material, you will need both equations When you are working with a maximum area of some type of border, you are usually working only with the area equation

Quadratic Word Problems Area Creating the Equation A = l x w (area) P = 2l + 2w (perimeter)

Quadratic Word Problems Area Creating the Equation Ex. Say you want to find equation showing the maximum enclosure area given a perimeter of 20m

Quadratic Word Problems Area Creating the Equation

Quadratic Word Problems Area Creating the Equation Ex. Say you want to find equation showing the maximum enclosure area given a perimeter of 20m P = 2l + 2w 20 = 2l + 2w (substitute 20 in for P) 10 = l + w (divide both sides by 2) l = 10 – w (isolate l)

Quadratic Word Problems Area Creating the Equation Ex. Say you want to find equation showing the maximum enclosure area given a perimeter of 20m Now, substitute l = 10 – w into A = l x w A = (10 – w)(w) (factored form) A = 10w - w² (standard form) A = -w² + 10w ( standard form)

Quadratic Word Problems Area Creating the Equation

Quadratic Word Problems Area Creating the Equation A = -w² + 10w 0 = - w² + 10w 0 = -w(w + 10) -w = 0 and w + 10 = 0 Zeros are 0 and -10 wv = -5 (take average of zeros) A = -(5)²+10(5) = 25 So, a width of 5m results in the maximum area of 25m²

Quadratic Word Problems Area Creating the Equation Mr. and Mrs. B want to install a rectangular swimming pool measuring 10m x 20m. They want to put a deck of uniform width around the pool. They have budgeted $1,920 to spend on the deck and know that construction and material costs are $30/metre

Quadratic Word Problems Area Creating the Equation Here we are using the area equation only, but we have to consider the two areas – that of the pool & that of the deck surrounding it Let’s let x represent the width of the uniform deck surrounding the pool Then, let’s state what the “new” overall length and width are of the pool & the deck together

Quadratic Word Problems Area Creating the Equation

Quadratic Word Problems Area Creating the Equation

Quadratic Word Problems Area Creating the Equation We know the inside area is: A = 10 x 20 = 200m² For the outside area we have to use other information given to determine it. We know that the B.’s have budgeted $1,920 to spend on the deck and that construction & material costs are $30 per metre. How can we get an area out of this?

Quadratic Word Problems Area Creating the Equation We know that the B.’s have budgeted $1,920 to spend on the deck and that construction & material costs are $30 per metre. How can we get an area out of this? Divide $1920 by $30 = 64 So, the total area of the pool and the deck will be 264 m²

Quadratic Word Problems Area Creating the Equation So, what is our equation? A = l x w 264 = (20 + 2x)(10 + 2x) 264 = 200 + 60x + 4x² 0 = 4x² + 60x + 200 – 264 0 = 4x² + 60x – 64 0 = 4(x² + 15x – 16) 0 = 4(x + 16)(x – 1) x = -16 or x = 1 (reject -16) so, the width of the deck should be 1m

Quadratic Word Problems Area Creating the Equation

Quadratic Word Problems Integers Creating the Problem The key with integer word problems is the translation of words into algebra

Quadratic Word Problems Integers Creating the Problem Consecutive numbers For consecutive numbers, you have to define x as any integer and the next number would be x + 1 (ex. If x was one, x + 1 = 2)

Quadratic Word Problems Integers Creating the Problem Consecutive odd numbers For consecutive odd numbers, you have to define x as any odd integer and the next consecutive odd number would be x +2 (ex. If x was one, x + 2 = 3)

Quadratic Word Problems Integers Creating the Problem Consecutive even numbers For consecutive even numbers, you have to define x as any even integer and the next consecutive even number would be x +2 (ex. If x was -4, x + 2 = -2)

Quadratic Word Problems Integers Creating the Problem Sum of squares Means you have to “square” or multiply each term by itself and then add the sum of the squares Ex. The sum of the squares of 2 consecutive numbers…

Quadratic Word Problems Integers Creating the Problem Sum of squares Ex. The sum of the squares of 2 consecutive numbers… x and x + 1 Sum of squares would be: = (x)²+ (x + 1)² = x²+ (x² + x + x + 1) (square first bracket and apply foil to the second bracket) = 2x²+ 2x + 1 Typically then this would be set to a number and you have to bring that number to the left, subtract it from 1 and create a trinomial (which can be factored) Let’s do a full example now…

Quadratic Word Problems Integers Creating the Problem Ex. The sum of the squares of 2 consecutive odd integers is 34. Find the two integers

Quadratic Word Problems Integers Creating the Problem

Quadratic Word Problems Integers Creating the Problem Ex. The sum of the squares of 2 consecutive odd integers is 34. Find the two integers Let x be the 1st odd integer So, x + 2 would be the next odd integer Sum of the squares means we have to have an (x)² and an (x + 2)² But, we are told the sum of these has to = 34 (x)² + (x + 2)² = 34

Quadratic Word Problems Integers Creating the Problem (x)² + (x + 2)² = 34 x² + x² + 2x + 2x + 4 = 34 2 x² + 4x + 4 = 34 2 x² + 4x + 4 – 34 = 0 2 x² + 4x – 30 = 0 2( x² + 2x – 15) = 0 2 (x + 5)(x – 3) = 0 So, the two zeros are -5 and 3 (these are also the two possible integers that work)

Homework (Given the Equation) Handout