Math 0200

Math 0094

Review Test 2

Solutions:

1. A 35% hydrochloric acid solution is to be mixed with a 65% hydrochloric acid solution to create 20 ounces of a 40% solution. How much of each solution is needed to create the mixture?

Let x = the amount of 35% solution

	Amount	% acid as a decimal	Amount of Pure Acid
35% Solution	X	.35	.35x
65% Solution	20 – x	.65	.65(20-x)
Mixture	20	.4	8

The amount of acid remains constant so,

.35 x + .65(20 – x) = 8

.35x +13 - .65x = 8

-.3x = -5 Divide by -.3

x = 16 2/3 ounces of 35 % solution

20 – x = 3 1/3 ounces of 65% solution

2. The complement of an angle is 20 degrees less than one-half of the angles’ supplement. Find the angle?

Let x = the angle.

Then

90 – x is the complement and

180 – x is the supplement, so

90-x = ½ (180-x) –20

90 – x = 90 - .5x –20

90 – x = 70 - .5 x

20 = .5 x

x = 40 degrees. So the angle is 40 degrees. Does this check and make sense? YES

3. A six-foot tall man casts an eight-foot long shadow. At the same time a tree casts a 45 foot long shadow. How tall is the tree?

Set up a proportion. Let x = the height of the tree.

Cross Multiply and

8x = 270

x = 33.75 feet tall.

4. A circular swimming pool is to be surrounded by a three- foot wide concrete walkway. If the diameter of the pool is 45 feet, how many square feet of concrete will be needed to create the walkway?

The radius of the inner circle is 22.5 feet and the radius of the outer circle is 25.5 feet.

The area of the walkway is the area of the outer circle minus the area of the pool.

Or Approximately 452.39 square feet.

5. Graph the line 5x – 3 y = 12.

Find the x and y intercepts by letting y and x = 0 respectively.

(2.4, 0), (0, -4)

6. Find the equation of the vertical and horizontal lines passing through (3, -9). What are the respective slopes?

Vertical lines always have slopes which are undefined and are, x = the x-coordinate. So the vertical line passing through (3, -9) is x = 3 and its slope is undefined.

Horizontal lines always have slopes of 0 and are, y = the y-coordinate. So the horizontal line passing through (3, -9) is y = -9 and its slope is 0.

7. Find the x and y intercepts of the line 8x +9y =16. Graph the line.

See number 5. Always label x and y intercepts as ordered pairs!.

X –intercept: (2, 0)

Y-intercept: (0, 16/9)

8. Find the slope of the line through (-2, 8) and (4, 6).

9. Find the equation of the line through (-2, 8), and (4, 6).

Use either the point slope formula or the slope intercept formula. I like y = mx +b, so that is what I’ll use.

Substitute in m, from problem 8, and one of the points for x and y.

6=(-1/3)(4) + b

6 = -4/3 + b

18/3 + 4/3 = b

b = 22/3

the equation is

y = -1/3 x + 22/3

10. Find the equation of the line perpendicular to 3x + 4y = 12 through the point (6, 2).

Solve the equation for y which will allow us to find the slope of this line. We want the opposite reciprocal for the slope of the perpendicular. Then use that slope and the point to find the y-intercept as we did in problem 9.

4y = -3x +12

y = -3/4 x +3

m = -3/4 so the perpendicular slope is 4/3.

Y = m x + b

2 = 4/3(6) + b

2 = 8 + b

b = -6

11. Karl and Jim are going to swim across Cedar Lake. Karl gives Jim a two-minute head star. If Karl averages 70 meters per minute and Jim averages 56 meters per minute, How long will it take Karl to catch up to Jim? How far will they each have swum?

Let x = the amount of time that Jim swims.

	Rate	Time	Distance
Karl	70	x - 2	70(x - 2)
Jim	56	x	56x

56x = 70x – 140

-14x = -140

x = ten minutes.

Karl will catch up to Jim 10 minutes after Jim left or 8 minutes after Karl left. They will each have swum 560 meters.

12. A company makes $1200 when the sell 15 computers in a week. Another week they lose $100 when they sell 5 computers. Find a linear function which relates their profits in terms of the number of computers sold. How many do they need to sell to at least break even?

Finding a linear function implies that we want to find the equation of the line. So we need two points.

Let x = the number of computers and y = the profit.

Points are (15, 1200) and (5, -100). Find the slope.

M = (1200 – (-100))/(15 – 5)

M = 130, or they are making a profit of $130 on each computer.

Plug into y = mx +b.

1200 = 130(15) + b.

1200 = 1950 + b

b = -750, so the linear function is

f(x) = 130x – 750

13. The DNR catches and tags 50 deer in a wildlife reserve which they then release back into the wild. Two weeks later they catch 35 deer, 10 of which have tags. Approximate the number of deer in the reserve.

Set up a proportion:

Let x = the number of deer in the reserve.