I think it would be around 19.64 pounds of CO2
56km/h is the answer to Question 9
1.5 hours is the answer to Question 10
Leave a comment if you'd like a step-by-step explanation
Answer:
No.
Step-by-step explanation:
No, Bob is not correct.
The formula he's using is the following:
![A=\frac{1}{2} ab\sin(C)](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%20ab%5Csin%28C%29)
The important thing here is that the angle is between the two sides.
In the given triangle, 120 is not between 8 and 18. Therefore, using this formula will not be valid.
Either Bob needs to find the other side first or find the angle between 8 and 18.
Answer: ![-0.5(x^2+1)^5+2.5](https://tex.z-dn.net/?f=-0.5%28x%5E2%2B1%29%5E5%2B2.5)
This is the same as writing -0.5(x^2+1)^5 + 2.5
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Explanation:
We take the integral to find the original f(x) function. Another term for "integral" is "antiderivative", which basically is the inverse of a derivative. There's more to the story, but this is the basic idea.
Apply a u-substitution
u = x^2+1
du/dx = 2x
du = 2xdx
xdx = du/2 = 0.5du
So,
![\displaystyle \int f ' (x) dx\\\\\\\displaystyle \int -5x(x^2+1)^4 dx\\\\\\\displaystyle -5\int (x^2+1)^4 xdx\\\\\\\displaystyle -5\int u^4(0.5du)\\\\\\\displaystyle -5*0.5\int u^4du\\\\\\\displaystyle -2.5\int u^4du\\\\\\-2.5\left(\frac{1}{4+1}u^{4+1}+C\right) \ \text{ ... don't forget about the +C}\\\\\\-2.5\left(\frac{1}{5}u^{5}+C\right)\\\\\\-2.5\left(\frac{1}{5}(x^2+1)^{5}+C\right)\\\\\\-0.5(x^2+1)^{5}-2.5C\\\\\\](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20f%20%27%20%28x%29%20dx%5C%5C%5C%5C%5C%5C%5Cdisplaystyle%20%5Cint%20-5x%28x%5E2%2B1%29%5E4%20dx%5C%5C%5C%5C%5C%5C%5Cdisplaystyle%20-5%5Cint%20%28x%5E2%2B1%29%5E4%20xdx%5C%5C%5C%5C%5C%5C%5Cdisplaystyle%20-5%5Cint%20u%5E4%280.5du%29%5C%5C%5C%5C%5C%5C%5Cdisplaystyle%20-5%2A0.5%5Cint%20u%5E4du%5C%5C%5C%5C%5C%5C%5Cdisplaystyle%20-2.5%5Cint%20u%5E4du%5C%5C%5C%5C%5C%5C-2.5%5Cleft%28%5Cfrac%7B1%7D%7B4%2B1%7Du%5E%7B4%2B1%7D%2BC%5Cright%29%20%5C%20%5Ctext%7B%20...%20don%27t%20forget%20about%20the%20%2BC%7D%5C%5C%5C%5C%5C%5C-2.5%5Cleft%28%5Cfrac%7B1%7D%7B5%7Du%5E%7B5%7D%2BC%5Cright%29%5C%5C%5C%5C%5C%5C-2.5%5Cleft%28%5Cfrac%7B1%7D%7B5%7D%28x%5E2%2B1%29%5E%7B5%7D%2BC%5Cright%29%5C%5C%5C%5C%5C%5C-0.5%28x%5E2%2B1%29%5E%7B5%7D-2.5C%5C%5C%5C%5C%5C%5C)
This means that
![f(x) = -0.5(x^2+1)^{5}-2.5C\\\\\\](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E%7B5%7D-2.5C%5C%5C%5C%5C%5C%5C)
To determine the constant C, we plug in x = 0 and replace f(x) with 2.
This is directly from the fact that f(0) = 2.
![f(x) = -0.5(x^2+1)^{5}-2.5C\\\\f(0) = -0.5(0^2+1)^{5}-2.5C\\\\2 = -0.5-2.5C\\\\-2.5C = 2+0.5\\\\-2.5C = 2.5\\\\C = 2.5/(-2.5)\\\\C = -1](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E%7B5%7D-2.5C%5C%5C%5C%5Cf%280%29%20%3D%20-0.5%280%5E2%2B1%29%5E%7B5%7D-2.5C%5C%5C%5C%5C2%20%3D%20-0.5-2.5C%5C%5C%5C%5C-2.5C%20%3D%202%2B0.5%5C%5C%5C%5C-2.5C%20%3D%202.5%5C%5C%5C%5CC%20%3D%202.5%2F%28-2.5%29%5C%5C%5C%5CC%20%3D%20-1)
Therefore,
![f(x) = -0.5(x^2+1)^{5}-2.5C\\\\\\f(x) = -0.5(x^2+1)^{5}-2.5(-1)\\\\\\f(x) = -0.5(x^2+1)^{5}+2.5\\\\\\](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E%7B5%7D-2.5C%5C%5C%5C%5C%5C%5Cf%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E%7B5%7D-2.5%28-1%29%5C%5C%5C%5C%5C%5Cf%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E%7B5%7D%2B2.5%5C%5C%5C%5C%5C%5C)
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Checking the answer:
Plug in x = 0. You should find that f(0) = 2. I'll let you do these steps.
Apply the derivative and use the chain rule to find:
![f(x) = -0.5(x^2+1)^5+2.5\\\\f'(x) = 5*(-0.5)(x^2+1)^{5-1}*\frac{d}{dx}(x^2+1)\\\\f'(x) = 5*(-0.5)(x^2+1)^{4}*(2x)\\\\f'(x) = -5x(x^2+1)^{4}\\\\](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-0.5%28x%5E2%2B1%29%5E5%2B2.5%5C%5C%5C%5Cf%27%28x%29%20%3D%205%2A%28-0.5%29%28x%5E2%2B1%29%5E%7B5-1%7D%2A%5Cfrac%7Bd%7D%7Bdx%7D%28x%5E2%2B1%29%5C%5C%5C%5Cf%27%28x%29%20%3D%205%2A%28-0.5%29%28x%5E2%2B1%29%5E%7B4%7D%2A%282x%29%5C%5C%5C%5Cf%27%28x%29%20%3D%20-5x%28x%5E2%2B1%29%5E%7B4%7D%5C%5C%5C%5C)
The answer is fully confirmed. Another way you can confirm the answer is to use free tools like GeoGebra or WolframAlpha. I show an example of this in the screenshot below.
There are 2 ways to get 6 by withdrawing 3 counters:
1,2 and 3
and 2, 2 and 2.
Probability ( drawing 1 , 2 and 3 in that order) = 2/9 * 3/8 * 4/7 = 24/504 = 1/21
there are 6 ways to draw any combination of 1,2 and 3 so this gives
6 * 1/21 = 2/7
probability of drawing 2,2 and 2 = 3/9 * 2/8 * 1/7 = 1/84
so the answer is 2/7 + 1/84 = 25/84