MATH 1201-01 - AY2024-T2 Written Assignment Unit 5
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Jan 9, 2024
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MATH 1201-01 - AY2024-T2 Written Assignment Unit 5 Using the formula: A=P(1+ r/n)nt
where P is the principal amount, r is the annual interest rate (decimal), n is the number of times interest is compounded per year, and t is the time in years.
For simple interest, the future value (A) is given by: A=P(1+rt)
where r is the annual interest rate (decimal), and t is the time in years.
A(20) = 8500(1 + 0.0812/12)
12*20
A(20) = 8500 (1+ 0006677)
240
= 42922.27
A (20) = 42922.27
Deposit compounded monthly with simple interest:
A(t) = P(1 + r/n*nt)
A(t) = P(1+rt)
A(t) = 8500(1+0.0812*20)
A(t) = 8500*2.624
A(t) = 22304
Thus, there was an exponential increase.
2. Graph the function f(x) = 5(0.5)
-x
and its reflection about the line y = x on the same axis, and give the x – intercept of the reflection. Prove that a
x = e
xlna
a
x = e
xlna
y = 5 *0.5
-x
= 5e
-xln0.5 y/5 = e
-xln0.5
ln(y/5) = -xln0.5 –ln(y/5)/ln0.5 = x y – ln(x/5)/ln0.5
X intercept is a point where y = 0 so, f-1(x) = -ln (0.5) ln(5x)
0 = -ln (0.5) ln(5x)
Ln(5x) = 0
5x = e0
5x = 1
X = 5
ax=exlna
The reflection of f(x)=5(0.5)−x about the line y=x is f−1(x)=−ln(0.5)ln(5x) The reflection of f(x)=5(0.5) −x about the line y=x is f−1(x)=−ln (0.5) ln(5x)
Therefore, x – intercept of reflection is function is (5,0)
3. Answer:
800 = 1000 e ln0.5/703800000t
0.8 = e
ln0.5/703800000t
Ln0.8 = ln0.5/703800000t
703800000*ln0.8/ln 0.5 =703800000*0.322 = t
T = 226572993
Therefore, 20% of the uranium has decayed in 226,572,993 years
(b) Suppose T is the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life.
Answer: If the half-life of uranium is 703,800,000, then T would be 2*703,800,000 = 1,407,600,000. (c)Prove that K= \frac{ln0.5} {T}. What is K for the uranium-235? Show the steps of your reasoning.
K = ln0.5 / T K = ln0.5 / 1,407,600,000 K= 703,800,000 (which is the half- life of Uranium, thus k =ln0.5 / T = True
Reference:
Abramson, J. (2021).
Algebra and trigonometry
(2nd ed.). OpenStax, TX: Rice University. Retrieved from
https://openstax.org/details/books/algebra-and-trigonometry-2e
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