Week 11 – Precalc 11

During week 11, I had a rough start in understanding the concept of solving word problems with quadratic functions. After seeking help, I managed to recognize the values of the two numbers using substitution method and having a maximum or minimum value was crucial in solving each problem. A perfect example is on page 343, Question 5:

The sum of the length and width of a rectangle is 20 cm. Determine the dimensions that produce the maximum area.

We know that Area of a rectangle is the product of Length and Width (i.e. Length x Width). Next, we need to substitute the two sides by using only variable () and not two ( and )!

Let = Length of the rectangle.

We know that Length + Width = 20cm; so we can re-write this equation as + Width = 20.

To isolate the Width value, we can move the terms around and get the following: Width = 20 –

Now, we can use substitution in the Area formula:

Area =

Distribute and then use the Completing of the Squares:

Area =

Area =

Area =

To find a “perfect square” trinomial, we must divide the coefficient of the second term () by 2 and then square that value.

Write this value inside the bracket so that the perfect square can be completed. Also, write the same value again with a subtraction sign. Don’t forget to multiply that -100 by -1 (at the beginning of the equation) in order to stand alone in the Standard Form.

In this Standard Form, we notice that the coefficient is a negative number (-1) which means the vertex is the maximum point and the solved dimensions will produce the maximum area. Looking at the values of p and q, the coordinates of the vertex are 10 and 100. Using the coordinate (10), we can solve the length and width of the rectangle.

Let = Length of the rectangle; then,
Let Width = 20 – ; then, Width

In conclusion, length is 10 cm and width is 10 cm.

Another word problem using an inequality is from question 5 from Unit 4.9 “Solving Linear and Quadratic Inequalities Algebraically”.

What is the maximum number of rides can Zac go on at the water park if it costs $2.50 to rent a mat, $1.25 for each ride, and he has only $20.00? Write an inequality to model this problem and then solve the problem.

First, we use substitution and let = the number of rides.

Second, we write the inequality to represent all the information in the question.

From here, we solve the variable by isolating that variable to one side of the inequality and moving the rest of constants to the other side.

We must always remember to reverse the inequality symbol when dividing each side by a NEGATIVE number!

In conclusion, the maximum number of rides Zac can go on is 14 rides.