Exercise 19 Page 423

Practice makes perfect

a The function below represents the breaking strength z of a manila rope.

z=8900 d^2

Here, d is the diameter of the rope. Since diameter cannot be negative, the domain of the function is all real numbers greater than or equal to zero. Domain:d≥ 0 The function z is of the form y= ax^2, where a>0. Therefore, the range of the function is all real numbers greater than or equal to zero. Range:z≥ 0

b To draw the graph of the function, we will make a table of values. We will start with 0, since d≥ 0.

d	8900(d)^2	z=8900(d)^2
0	8900( 0)^2	0
1	8900( 1)^2	8900
2	8900( 2)^2	35 600
3	8900( 3)^2	80 100

Let's plot the points ( 0, 0), ( 1, 8900), ( 2, 35 600), and ( 3, 80 100) and draw a smooth curve through them.

c Let z_1 and d_1 be the breaking strength and diameter of a manila rope.

z_1=8900 d_1^2Let's multiply both sides of the equation by 4.

z_1=8900( d_1)^2

MultEqn

LHS * 4=RHS* 4

4z_1=8900(d_1)^2* 4

WritePow

Write as a power

4z_1=8900(d_1)^2* 2^2

ProdPow

a^m* b^m=(a * b)^m

4z_1=8900(2d_1)^2

This means that a manila rope with four times the breaking strength has two times the diameter of the weak one. This is because the relationship is quadratic. z_1=8900 d_1^2 ⇒ 4z_1=8900(2d_1)^2

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