Types of Word Problems:

1) Flight of an object

2) Revenue 

3) Triangles 

4) Consecutive integers 

5) Maximum area

 

 

Example 1: Flight of an object problem

 

The path of the soccer ball after it was thrown in the air is given by the following equation: h= -0.40d^2+2d+1.5 where h is the height and d is the horizontal distance in metres. 

 

 

a) What is the initial height of the soccer ball? [In the formula]

Therefore, the intial height of the soccer ball it 1.5

 

b) What is the maximum height reached by the soccer ball and what is the horizontal distance does this occur at? [Vertex]

*When it asks for the vertex, automatically your brain should know the option is completing the square*

 h= -0.40d^2+2d+1.5

h= (-0.40d^2+2d)+1.5

h= -0.40(d2-5)+1.5

h=-0.40((d2-5+6.25)-6.25)+1.5                    Therefore, the maximum height reached by the soccer ball

h=-0.40(d-2.5)^2+2.5+1.5                                 is 4m and the horizontal distance it occurs at is 2.5m 

h= -.40(d-2.5)^2+4

(2.5,4)

 

Example 2: Revenue problem

 

The environmental club sells sweatshirts as a fundraiser. They sell 1200 shirts a year at $20 each. They are planning to increase the price. A survey indicates that, for every $2 increase in price, there will be a drop of 60 sales a year. What should the selling price be in order to maximize the revenue. 

 

Step 1: Write the equation [Hint: Revenue= (Price)(Quanity)]

 

Price= 20+2x                                R= (20+2x)(1200-60x)

Quanity= 1200-60x

 

Step 2: Using the equation, expand and simplify to standard form

 

R= (20+2x)(1200-60x)

R= 24000-1200x+2400x+24000

R= -120x2-1200x+2400x+24000

R= -120x2+1200x+24000

 

Step 3: Since this question is not asking for dimentsions, we know not to use the quadratic formula. However, since its mentioning maximize, in our heads we should automatically know its asking for vertex. In this case we can use many methods like factoring. finding zeros, axis of summetry, but we are using completing the square because its much faster and efficient.

 

R= (-120x2+1200x)+24000

R= -120(x2-10x)+24000

R= -120((x2-10+25)-25)+24000

R= -120(x2-5)^2+3000+24000

R= -120(x-5)^2+27000

(5,27000)

 

Step 4: Sub 5 into the equation of price equation

 

P= (20+2x)

P= 20+2(5)

P= 20+10                                                   Therefore, the selling price should be $30 in order to maximize the revenue

P =30 

 

 

 

 

 

 

Example 3: Triangle problem

 

The length of one leg of a right triangle is 1cm more than that of the other leg. The lengh of the hypotenuse is 9 cm more than double that of the shorter leg. Find the lengths of all three sides of the triangle[Draw a diagram] 

 

 

Recall: Formula for pythagorean theorem

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 3: Since you are finding the lenghs of the three sides, you

                      find the quadratic formula

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 4: Consective integers problem

 

The product of two consective numbers is 3308. What are the numbers?

 Let x represent the 1st integer

 Let x+1 represent the 2nd integer

 

Step 1: Expand the equation      (x)(x+1)=3308

                                                       x2+x=3308

                                                        x2+1-3308=0 

 

Step 2: Use the quadratic formula

to find the two possible integers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 5: Maximum area problem

 

The rectangular dock measures sides 5m by 4m. A new dock is going to be made by increasing each side by the same amount. The area of the new patio is going to be 42cm2. What are the dimensions of the new patio?