Select three ratios that are equivalent to 11 : 1
1 answer:
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Answer:
See below.
Step-by-step explanation:
1)
For Ben's social media post, we are told that it was initially shared to only 2 people. However, each of those shares it with 3 people. In other words, Ben's post triples every day. This means that the rate of growth is 3. Thus, Ben's post as an exponential function is:
The 2 represents the initial amount at day 0, and the 3 represents the rate at which the post grows each day.
2)
Similarly, Carter's post was originally sent to 10 people. However, each of the 10 shares it with two others. Therefore, it doubles each day. Thus, the rate is 2. So, Carter's exponential function is:
The 10 represents the initial amount while the 2 represents the rate.
3)
I've use Desmos to graph the functions. Refer to the picture attached. Red is Amber, blue is Ben, and green is Carter.
4)
To find out the number of shares each student's post will receive on day 3 and day 10, simply plug the numbers into the functions. Therefore:
For Amber:
Day 3:
Day 10:
For Ben:
Day 3:
Day 10:
And for Carter:
Day 3:
Day 10:
5)
Amber's new graph of:
will be a shifted version of her original graph. The new graph will be shifted upwards by 45 units. However, since the 45 is not affected by the x in any way, the change will be minuscule. For instance, her number of shares on day 10 of the old graph is 3,145,728. Even with the additional 45, it will just be: 3,145,773. With enough days, the additional number becomes trivial.
6)
On the 10th day, Amber's post was shared almost 3 million times, while Ben's was shared "only" 118,098 times and Carter's with 10,240. Undoubtedly, Amber's post was the fastest.
7)
If you want to most shares as possible, choose Amber with fewer friends initially but more shares. This is because the exponential potential of a number even just a bit higher is insane. For instance, 3^10 (Ben's rate at 10th day) is only 59,049 while Amber's rate will be 4^10 which is 1,048,576. Amber's number is almost 18 times higher than Ben's. Even if Ben's initial number was say 1,000 or even 1,000,000, Amber would catch up rather quickly. Therefore, the rate is the quintessential part of an exponential growth function.
The time value of money calculation can be performed using formula equations or online calculators.
The correct responses are;
- 1) Option 3
- 2) Option 2
- 3) The difference in principal is approximately $8,000
- The difference in interest earned is approximately $2,977.87
- 4) It is better to invest more money at the beginning of the 30 years
Reasons:
Option 1: Present value = 0
Amount invested per month, A = $25/month
The Annual Percentage Rate, APR, r = 3.25%
Number of years = 30
The future value of an annuity is given by the formula;
In option 1, m = 12 periods per year
Therefore;
Contribution = $25 × 12 × 30 = $9,000
Total interest earned = $15,209.3 - $9,000 = $6,209.3
Final balance = $15,209.3
Option 2: Present value = 0
Amount, A = $75/quarter
m = 4 periods per year
The Annual Percentage Rate, APR = 4.00%
Therefore;
The effective interest rate is therefore;
Using an online calculator, FV = $17,467.04
Contribution = $75 × 4 × 30 = $9,000
Total interest earned = $17,467.04 - $9,000 = $8,467.04
Final balance = $17,467.04
Option 3: Present value = $1,000
APR = 6.25%
m = 12 period per year
Number of years, t = 30 years
Therefore;
Contribution = $1,000
Total interest earned = $6,489.17 - $1,000 = $5,489.17
Final balance = $6,489.17
The table of values is therefore;
1) The option that has the least amount invested are <u>option 3</u>
Option 3 investment plan is a present value of $1,000, invested for 30 years at 6.25% APR compounded monthly.
2) <u>Option 2</u> yielded the highest amount at the end of 30 years, given that the APR is higher than the APR for option 1, although the amount invested over the period are the same.
The basis of option 2 investment plan is $75 invested quarterly at 4.00% APR compounded monthly for 30 years.
3) The difference in the principal invested for the highest and lowest final balance is $9,000 - $1,000 = <u>$8,000</u>
The difference in the interest earned is; $8,467.04 - $5,489.17 = <u>$2,977.87</u>
4) In option 1 the present value is zero, therefore zero amount was invested at the beginning.
The interest to investment ration is 6,209.3:9,000 ≈ 0.7:1
In option 3, all the money was invested at the beginning.
The interest to investment ratio of option 3 is; 5,489.17:1,000 ≈ 5.5:1
Given that the interest to investment ratio, which is the return on investment is larger when more money is saved at the beginning as in option 3, <u>it is better to invest more money at the beginning</u>.
Learn more about future value of an annuity here: