The goal of this problem is to use the definition of the derivative to find ah + b where a and b are constants. find these const
1 answer:
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Answer:
(a) n(t) = P(0)*e^(0.010939940t)
(b) 12,674,681 (nearest unit)
(c) 14 years (nearest year)
Step-by-step explanation:
rate = 1.1% / year = 1.011
(a)
P(0) = 12,000,000 = population in 2010
In compound interest format, after t years
P(t) = P(0)* (1.011)^t
Given format = P(0)* e^(rt)
therefore
e^(rt) = 1.011^t use law of exponents
(e^r)^t = 1.011^t
e^r = 1.011
r = log_e(1.011) = 0.010939940 (to 9 decimal places)
required formula is
n(t) = P(0)*e^(0.010939940t)
(b)
in 2015,
P(0)=12000000, n = 5 (years after 2010)
n(5) = 12000000*e^( 0.010939940 * 5 ) = 12,674,680.6 = 12,674,681 (nearest unit)
(c)
to reach 14 million, we equate
n(t) = 14,000,000
12,000,000 *e^(0.010939940*t) = 14,000,000
e^(0.010939940*t) = 14000000/12000000 = 7/6
take log on both sides
0.010939940*t = log(7/6)
t = log(7/6) / 0.010939940 = 14.091 years = 14 years to the nearest year.
See graph attached. Y-axis is in millions, x-axis is in years.
The first equation given
We have to add 1 and 5 to the right side first. We will get,
To get rid of 6 from the right side we have to subtract 6 from both sides.
To find n we have to move -2 to the other side by dividing both side by -2.
So we have got the required answer for the first question.
The solution is n = 0.
The second equation given,
First we have to move 7x to the left side by subtracting it from both sides.
Now we have to move -2 to the right side by adding 2 to both sides.
We have got the required answer for the second question.
The solution is x = -7.
The third equation given,
We have to get rid of that negative sign from both sides. As we have negative sign to both sides we can cancel it out. We will get,
Now we have to move 4 to left side by subtracting it from both sides.
So we have got the required answer .
The solution is x = 4.