Math 103. Homework # 1. Solutions

1. Find the minimum cost C (in dollars), given that

5(C-25) ³ 1.75 +2.5 C

Solution. First, simplify this inequality:

5C-125 ³ 1.75 +2.5 C

5C -2.5 C ³ 125 + 1.75

C ³ 126.75 2.5 =50.7.

So the minimum cost is 50.7

2. Determine whether each of the following statements is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.

(a) |a-b|=|b-a|
(b) Ö[(a² - b²)]=|a|-|b|

Solution. Statement (a) is true, the reason being that the absolute value of a number x is the same as the absolute value of -x. Thus |a-b|=|-(a-b)|=|b-a|.

Statement (b) is false. For instance, if a=5 and b=4, then

_____ Ö25-16   = 3 ¹ |5|-|4| = 1.

3. The distribution of income in a certain city can be described by the exponential model y=(2.8) ( 10¹¹)(x)^-1.5, where y is the number of families with an income of x or more dollars. How many families in this city have an income of $20,000 or more?

Solution. Plugging the value x=20,000 = 2 (10⁴) into the equation given and simplify. The details are

y = (2.8 ( 1011)) (2 (104))-3/2 = 2.8 Ö8 1011-6 = (0.98994949)(105)=98994.949

So 98,994 families have an income of $20,000 or more.

4. Simplify each of the following expressions.

(a)




(4x2)(x2+4)(2x+8)-(x2+8x-4)(4x3)


(b)




6(2x+1)2(x2+x)1/2 + (2x+1)4 2(x2+x)1/2

Solution. (a) First multiply: 8x⁵ +32 x³ +32 x⁴+128 x² -( 4x⁵ + 32 x⁴ -16 x³), then collect equal powers of x:

(8- 4)x5 +(32-32)x4+(32+16)x3 +128 x2 = 4x5 +48x3 +128x2.

(b) Add the same way that you would add fractions (common denominator), then simplify:

6(2x+1)2(x2+x)1/2 + (2x+1)4 2(x2+x)1/2 = 6(2x+1)2 (x2+x)1/2 2(x2+x)1/2+(2x+1)4 2(x2+x)1/2 = 12 (2x+1)2(x2+x) +(2x+1)4 2(x2+x)1/2 = (2x+1)2 ( 12(x2+x) +(2x+1)2) 2(x2+x)1/2 = (2x+1)2(16x2+16x+1) 2(x2+x)1/2

5. A furniture store offers free setup and delivery charges to all points within a 25 mi radius of its warehouse distribution center. If you live 20 mi east and 14 mi south of the warehouse, will you incur a delivery charge? Justify your answer.

Solution. Place the store at the origin of a coordinate system. Your house is then located at the point (20,-14), and the distance to the origin is Ö[(20²+(-14)²)]=24.413111, which is less than 25. Thus there are no delivery charges.