Question

1. A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years? What if the deposit was calculated using simple interest? Could you see the situation in a graph?

2. Graph the function  and its reflection about the line y=x on the same axis, and give the x-intercept of the reflection. Prove that . [Suggestion: type  {- 7 < x < 2} {0 < y < 7} in desmos, and then type its inverse function.]

3. How long will it take before twenty percent of our 1,000-gram sample of uranium-235 has decayed? [See Section 6.6 Example 13]

The decay equation is , where t is the time for the decay, and K is the characteristic of the material. 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. Prove that  . What is K for the uranium-235?

Answer 1

1. We should use the compound interest to calculate the account in 20 years.

$$S = A(1 + P/100\%)^n=$$

$$=8500*(1+8.12\%/100\%)^{240}=$$

$$=8500*(1+0.0812)^{240}=$$

$$=8500*137228844=1166445174000(\$)$$

 Where S is the cost of the account in 20 yeras, A - the initial cost, P - monthly interest, n - amount of the settlement periods,

$$n= 20 years * 12 months = 240 months.$$

If the deposit was calculated using simple interest, we should use the next formula:

$$S=A*(1+P*n)=8500*(1+0.0812*240)=174148 (\$)$$

2 To get the reflection about the function y=x, we should express the function as the inverse one relatively to x. For example, if we have the function y=5x, the reflection will be y=x/5. We can show it on the picture and get the point of the interseption with the X axis, it is the point (0,0) when the x=0, and so y=5x=5*0=0, and on the picture we can see the interception of the line in this point, we can use this information as a proof.

3 The uranium decay equation can be written by the following formula:

$$m=m_0*e^{-Lt},$$

where m0 is the initial mass of the uranium, m - the mass of the 80% of uranium when the 20% of it is decayed, L- the decay constant of uranium, maybe it is K of the task requirements, t - the decay period. So we should solve the exponential equation:

$$1/e^{Lt}=m/m_0$$

$$e^{Lt}=m_0/m=100\%(m)/80\%(m)=1.25$$

$$Lt=ln(1.25)$$

$$t=ln(1.25)/L=ln(1.25)*T(1/2)/0.693$$

$$L=0.693/T(1/2),$$

where T is the half-decay period of the uranium-235, 700 million years.

$$t=0.223*700,000,000/0.693=225397526 years$$