What is the solution to the equation? round your answer to two decimal places. 6•e^x=25
1 answer:
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Answer:
(a) Probability that Y falls into the dangerous region is 0.0013.
(b) Probability that the mean Y-bar falls into the dangerous region is 0.00001.
Step-by-step explanation:
We are given that a certain capsule is manufactured so that the dosage of the active ingredient follows the distribution Y ~ N(μ = 10 mg, σ = 1 mg).
A dosage of 13 mg is considered dangerous.
Let Y = <u><em>dosage of the active ingredient </em></u>
The z-score probability distribution for normal distribution is given by;
Z = ~ N(0,1)
where, = population mean = 10 mg
= standard deviation = 1 mg
(a) Probability that Y falls into the dangerous region is given by = P(Y 13 mg)
P(Y 13 mg) = P(
) = P(Z
3) = 1 - P(Z < 3)
= 1 - 0.9987 = <u>0.0013</u>
The above probability is calculated by looking at the value of x = 3 in the z table which has an area of 0.9987.
(b) We are given that a dosage of 13 mg is considered dangerous. And we sample 49 capsules at random.
Let = sample mean dosage
The z-score probability distribution for sample mean is given by;
Z = ~ N(0,1)
where, = population mean = 10 mg
= standard deviation = 1 mg
n = sample of capsules = 49
So, Probability that the mean Y-bar falls into the dangerous region is given by = P(
13 mg)
P(Y 13 mg) = P(
) = P(Z
21) = 1 - P(Z < 21)
= <u>0.00001</u>
First solve x³=9x
We can ignore x = -3 because it is not in the first quadrant.
<span>So our integral is going to go from x=0 to x=3.
</span>
<span>Now we can use the formula
</span><span>
</span>![\int\limits_{0}^{3}(\pi (9x)^2-\pi(x^3)^2)dx \\\int\limits_{0}^{3}(\pi (9x)^2-\pi(x^3)^2)dx \\\pi\int\limits_{0}^{3}(81x^2-x^6)dx \\2\times \int\limits_{0}^{3}[\pi(9x)^2-\pi(x^3)^2]dx \\\pi\left[81\frac{x^3}{3}-\frac{x^7}{7}\right]_{0}^{3} \\ \\\frac{2916\pi}{7}](https://tex.z-dn.net/?f=%5Cint%5Climits_%7B0%7D%5E%7B3%7D%28%5Cpi%20%289x%29%5E2-%5Cpi%28x%5E3%29%5E2%29dx%0A%5C%5C%5Cint%5Climits_%7B0%7D%5E%7B3%7D%28%5Cpi%20%289x%29%5E2-%5Cpi%28x%5E3%29%5E2%29dx%0A%5C%5C%5Cpi%5Cint%5Climits_%7B0%7D%5E%7B3%7D%2881x%5E2-x%5E6%29dx%0A%5C%5C2%5Ctimes%20%5Cint%5Climits_%7B0%7D%5E%7B3%7D%5B%5Cpi%289x%29%5E2-%5Cpi%28x%5E3%29%5E2%5Ddx%0A%5C%5C%5Cpi%5Cleft%5B81%5Cfrac%7Bx%5E3%7D%7B3%7D-%5Cfrac%7Bx%5E7%7D%7B7%7D%5Cright%5D_%7B0%7D%5E%7B3%7D%20%5C%5C%20%5C%5C%5Cfrac%7B2916%5Cpi%7D%7B7%7D)
<span>
</span>
We can ignore x = -3 because it is not in the first quadrant.
<span>So our integral is going to go from x=0 to x=3.
</span>
<span>Now we can use the formula
</span><span>
</span>
</span>