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3 years ago

Round each fraction to the nearest whole, 1/2, or 0:

Mathematics

1 answer:

Juliette [100K]3 years ago

5 0

Answer:

1 2/5 = 1 1/2

7/9 = 1

4/9 = 1/2

3 5/7 = 4

Step-by-step explanation:

1. we must round 1 2/5

based on the answer choices we can determine that 1 1/2 is the most rational answer. This is because it is not only the only answer with the whole number, 1 but also 2/5 would round to 1/2.

2. We must round 7/9

This would round up to 1. This is because it is closest to the whole number, not 0 and not 1/2.

7 * 2 = 14

9 * 2 = 18

14/18 is only 4/18 away from being 1. This means that it would round up to 1.

3. we must round 4/9

This rounds to 1/2. If you multiply both side by 2:

4 * 2 = 8

9 * 2 = 18

8/18 for that to be 1/2 it would need 1/18. Thus this rounds up to 1/2.

4. We must round 3 5/7

5/7 would round up to a whole number.

3 + 1 = 4

Thus 3 5/7 = 4 when rounded to the nearest whole number.

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To be clear, the given relation between time and female population is an integral:
 $t = \int { \frac{P+S}{P[(r - 1)P - S]} } \,
dP$

The problem says that r = 1.2 and S = 400, therefore substituting:
 $t = \int { \frac{P+400}{P[(1.2 - 1)P - 400]}
} \, dP$ 

= $
\int { \frac{P+400}{P(0.2P - 400)} } \, dP$

In order to evaluate this integral, we need to write this rational function in a simpler way:
 $\frac{P+400}{P(0.2P - 400)} = \frac{A}{P} +
\frac{B}{(0.2P - 400)}$ 

where we need to evaluate A and B. In order to do so, let's calculate the LCD:
 $\frac{P+400}{P(0.2P - 400)} = \frac{A(0.2P -
400)}{P(0.2P - 400)} + \frac{BP}{P(0.2P - 400)}$ 

the denominators cancel out and we get:
P + 400 = 0.2AP - 400A + BP
 = P(0.2A + B) - 400A

The two sides must be equal to each other, bringing the system:
 $\left \{ {{0.2A + B = 1} \atop {-400A =
400}} \right.$ 

Which can be easily solved:
 $\left \{ {{B=1.2} \atop {A=-1}} \right.$ 

Therefore, our integral can be written as:
 $t = \int { \frac{P+400}{P(0.2P - 400)} } \,
dP = \int {( \frac{-1}{P} + \frac{1.2}{0.2P-400} )} \, dP$ 
= $- \int { \frac{1}{P} \, dP +
1.2\int { \frac{1}{0.2P-400} } \, dP$ 
= $- \int { \frac{1}{P} \, dP +
6\int { \frac{0.2}{0.2P-400} } \, dP$ 
= - ln |P| + 6 ln |0.2P - 400| + C

Now, let’s evaluate C by considering that at t = 0 P = 10000:
0 = - ln |10000| + 6 ln |0.2(10000) - 400| + C
C = ln |10000| - 6 ln |1600|
C = ln (10⁴) - 6 ln (2⁶·5²)
C = 4 ln (10) - 36 ln (2) - 12 ln (5) 
 Therefore, the equation relating female population with time requested is:
t = - ln |P| + 6 ln |0.2P - 400| + 4 ln (10) - 36 ln (2) - 12 ln (5)

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