Answer:
![(a)\ (1 + 3x)^6 = 728x^6 + 1458x^5 + 1215x^4 + 540x^3 + 135x^2 + 18x + 1](https://tex.z-dn.net/?f=%28a%29%5C%20%281%20%2B%203x%29%5E6%20%3D%20728x%5E6%20%2B%201458x%5E5%20%2B%201215x%5E4%20%2B%20540x%5E3%20%2B%20135x%5E2%20%2B%2018x%20%2B%201)
![(b)\ (1 + 3(-0.01))^6 = 0.8330](https://tex.z-dn.net/?f=%28b%29%5C%20%281%20%2B%203%28-0.01%29%29%5E6%20%3D%200.8330)
Step-by-step explanation:
Given
![(1 + 3x)^6](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6)
Solving (a): Expand
To do this, we make use of pascal triangle
When the exponent is 6, the factors are: 1, 6, 15, 20, 15, 6 and 1
As a rule, the exponent of 1 will start from 0 and then increase by 1 in each term while the exponent of 3x will start from 6 and then decrease by 1 in each term.
So, we have:
![(1 + 3x)^6 = 1 * 1^0 * (3x)^6 + 6 * 1^1 * (3x)^5 + 15 * 1^2 * (3x)^4 + 20 * 1^3 * (3x)^3 + 15 * 1^4 * (3x)^2 + 6 * 1^5 * (3x)^1 + 1 * 1^6 * (3x)^0](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6%20%3D%201%20%2A%201%5E0%20%2A%20%283x%29%5E6%20%2B%206%20%2A%201%5E1%20%2A%20%283x%29%5E5%20%2B%2015%20%2A%201%5E2%20%2A%20%283x%29%5E4%20%2B%2020%20%2A%201%5E3%20%2A%20%283x%29%5E3%20%2B%2015%20%2A%201%5E4%20%2A%20%283x%29%5E2%20%2B%206%20%2A%201%5E5%20%2A%20%283x%29%5E1%20%2B%201%20%2A%201%5E6%20%2A%20%283x%29%5E0)
Expand
![(1 + 3x)^6 = 1 * (3x)^6 + 6 * (3x)^5 + 15 * (3x)^4 + 20 * (3x)^3 + 15 * (3x)^2 + 6 * (3x)^1 + 1 * (3x)^0](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6%20%3D%201%20%2A%20%283x%29%5E6%20%2B%206%20%2A%20%20%283x%29%5E5%20%2B%2015%20%20%2A%20%283x%29%5E4%20%2B%2020%20%2A%20%20%283x%29%5E3%20%2B%2015%20%2A%20%20%283x%29%5E2%20%2B%206%20%20%2A%20%283x%29%5E1%20%2B%201%20%2A%20%283x%29%5E0)
![(1 + 3x)^6 = 1 * 728x^6 + 6 * 243x^5 + 15 * 81x^4 + 20 * 27x^3 + 15 * 9x^2 + 6 * 3x + 1](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6%20%3D%201%20%2A%20728x%5E6%20%2B%206%20%2A%20%20243x%5E5%20%2B%2015%20%20%2A%2081x%5E4%20%2B%2020%20%2A%20%2027x%5E3%20%2B%2015%20%2A%20%209x%5E2%20%2B%206%20%20%2A%203x%20%2B%201)
![(1 + 3x)^6 = 728x^6 + 1458x^5 + 1215x^4 + 540x^3 + 135x^2 + 18x + 1](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6%20%3D%20728x%5E6%20%2B%201458x%5E5%20%2B%201215x%5E4%20%2B%20540x%5E3%20%2B%20135x%5E2%20%2B%2018x%20%2B%201)
Solving (b): (0.97)^6
Rewrite as:
![(0.97)^6 = (1 - 0.03)^6](https://tex.z-dn.net/?f=%280.97%29%5E6%20%3D%20%281%20-%200.03%29%5E6)
Express -0.03 as 3 * -0.01
![(0.97)^6 = (1 + 3(-0.01))^6](https://tex.z-dn.net/?f=%280.97%29%5E6%20%3D%20%281%20%2B%203%28-0.01%29%29%5E6)
So, by comparing:
and ![(1 + 3(-0.01))^6](https://tex.z-dn.net/?f=%281%20%2B%203%28-0.01%29%29%5E6)
![x = -0.01](https://tex.z-dn.net/?f=x%20%3D%20-0.01)
Recall that:
![(1 + 3x)^6 = 728x^6 + 1458x^5 + 1215x^4 + 540x^3 + 135x^2 + 18x + 1](https://tex.z-dn.net/?f=%281%20%2B%203x%29%5E6%20%3D%20728x%5E6%20%2B%201458x%5E5%20%2B%201215x%5E4%20%2B%20540x%5E3%20%2B%20135x%5E2%20%2B%2018x%20%2B%201)
This gives:
![(1 + 3(-0.01))^6 = 728(-0.01)^6 + 1458(-0.01)^5 + 1215(-0.01)^4 + 540(-0.01)^3 + 135(-0.01)^2 + 18(-0.01) + 1](https://tex.z-dn.net/?f=%281%20%2B%203%28-0.01%29%29%5E6%20%3D%20728%28-0.01%29%5E6%20%2B%201458%28-0.01%29%5E5%20%2B%201215%28-0.01%29%5E4%20%2B%20540%28-0.01%29%5E3%20%2B%20135%28-0.01%29%5E2%20%2B%2018%28-0.01%29%20%2B%201)
Using a calculator
![(1 + 3(-0.01))^6 = 0.000000000728 -0.0000001458+ 0.00001215 -0.00054 + 0.0135 -0.18 + 1](https://tex.z-dn.net/?f=%281%20%2B%203%28-0.01%29%29%5E6%20%3D%200.000000000728%20-0.0000001458%2B%200.00001215%20%20-0.00054%20%20%2B%200.0135%20-0.18%20%2B%201)
![(1 + 3(-0.01))^6 = 0.83297200492](https://tex.z-dn.net/?f=%281%20%2B%203%28-0.01%29%29%5E6%20%3D%200.83297200492)
--- approximated
Answer:
You would plot the following points, remembering that they’re in the form (x,y) = (time, height):
(1,35) (2,60) (3,92) (4,120) (5,175)
The ordered pairs would be closer together, and there would be 4 times more of them if you were to measure every 15 seconds instead. You would have a line with more points that were closer together.