Math
Exercise 8
Choose the best answer. Solve the following systems:
Question 1:
A natural number increased by 5 is equal to 36 times the reciprocal of the number. Find the number. Select the correct answer.
(a) |
1 | (b) |
4 |
(c) |
9 | (d) |
31 |
(b) |
4 |
|
x + 5 = 36/x, so x2 + 5x -36 = 0 Use the quadratic formula and find x = -9 and x = 4. Since x is a
natural number, x=4. |
Question 2:
Two cyclists, starting together, travel 75 km each along the same course. One of them travels 5 km/h slower than the other and covers the distance in 30 minutes longer time. The speed (in km/hr) of the slowest cyclist is:
(a) |
12 | (b) |
15 |
(c) |
24 | (d) |
29 |
(c) |
24 |
|
d= 75 km A: 75/y = x So, integrating these two formulas gives us: 5y2 - 2.5y – 37.5 = 0 Using the quadratic formula gives us two solutions for y: -3 (is not a usable solution, since we are looking for a positive number) and 2.5. So the speed for A is 75/2.5 = 30 and for B: 75/3 = 25 km/hr. |
Question 3:
A swimming pool is supplied with water through two pipes. When both pipes are used the pool can be filled in 4.8 hours. The larger pipe alone will fill the pool in 4 hours less time than the smaller one. Which of the following numbers is closest to the time (in hours) it takes the small pipe to fill the pool?
(a) |
1 | (b) |
3 |
(c) |
5 | (d) |
10 |
(d) |
10 |
|
Content swimming pool = l x b x h = C
Flow (Big pipe) + flow (Small pipe) = C/4.8 Integrating equation 2 and 3 in eq.1 gives: .2 t2 - 2.8t + 4 = 0 Use the quadratic formula and get 12.4 as an answer. |
Question 4:
In a number of two digits the tens digit is greater than the units digit by 2. If the digits are reversed, the product of the new number and the original number is 403. Which of the following numbers the original number?
(a) |
23 | (b) |
31 |
(c) |
45 | (d) |
51 |
(b) |
31 |
|
Let two digit number = ab. {[(b + 2) x 10] + b}{(b x 10) + ( b+ 2)} = 403, Now use the quadratic formula, which finds b = 1 and b = -3 (not usable)
|
Question 5:
The area of a rectangle remains unaltered when the length is increased by 4 m and the width diminished by 3 m. If instead the length is increased by 16 m and the width diminished by 10 m, the area is diminished by one-third. The original length (in metres) of the rectangle is:
(a) |
10 | (b) |
13 |
(c) |
16 | (d) |
22 |
(c) |
16 |
|
Area=length times width = (l + 4)(w-3) Take w out of the first equation, width = 3/4x + 3 |
Question 6:
The number of bacteria in a certain culture was observed to double every 24 hours. Assume that there were 105 bacteria present at the first count.
i) The number of bacteria after 4 days would be:
(a) |
420 | (b) |
525 |
(c) |
840 | (d) |
1680 |
(d) |
1680 |
|
after one day: 105 X 2 =
210 after two days: 210 X 2 = 420 after three days: 420 X 2 = 840 after four days: 840 X 2 = 1680 |
ii) The number of bacteria after t days would be:
(a) |
105t | (b) |
105 ∙ 2t |
(c) |
105t | (d) |
105 ∙ 2t |
(b) |
105 ∙ 2t |
|
two days: (105 + 105)(105 + 105) and so on…, so |
iii) Approximately how many bacteria were present after 1 ½ days. Choose the option closest to the correct answer.
(a) |
260 | (b) |
300 |
(c) |
315 | (d) |
320 |
(b) |
300 |
|
after 1.5 days: 105 x 21.5 = 297, which is closest to 300. |
Question 7:
The carbohydrate, fat, and protein content in ounces, of 1 ounce of milk, meat, and vegetables are given in the table. Find the amount of each food necessary for a meal of 3.4 ounces of carbohydrate, 3.2 ounces of fat and 4.9 ounces of protein.
Milk |
Meat |
Vegetable |
|
Carbohydrates |
0.1 |
0 |
0.6 |
Fat |
0.2 |
0.3 |
0 |
Protein |
0.1 |
0.5 |
0.1 |
The amount of milk (in ounces) in the answer is:
(a) |
8 |
(b) |
6 |
(c) |
5 |
(d) |
4 |
(d) |
4 | |||||||||
|
The necessary amounts of each are built up out of all three food products:
Procedure (in this order, do it step by step!): You will now have: c = 5, b = 8 and a = 4 |
Question 8:
In 1990, the Intergovernmental Panel on Climate Change predicted that the average temperature on the earth would rise 0.3oC per decade in the absence of international controls on greenhouse emissions (Science News, June 23, 1990, p. 391). In 1970 the average global temperature was 15oC.
i) Determine a linear equation giving the average global temperature (T) in degrees Celsius in terms of t, the number of years since 1970.
(a) |
T = 0.3t + 15 | (b) |
T = 1970t + 0.3 |
(c) |
T = 0.03t + 15 | (d) |
T = 15t + 0.3 |
(a) |
T = 0.03t + 15 |
|
We started with a temperature of 15 degrees, adding 0.3 degrees per decade. |
ii) Scientists have estimated that the sea level will rise by 65 cm if the average global temperature rises to 19oC. When will this occur? Choose the option closest to the correct answer.
(a) |
1972 | (b) |
1983 |
(c) |
2000 | (d) |
2100 |
(d) |
2100 |
|
When will T= 19 degrees? |
