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KATRIN_1 [288]
3 years ago
13

What is the sum of the series?

Mathematics
1 answer:
marta [7]3 years ago
8 0
t_n = a r^{n-1}

a = 4, r = 3, t_n = 8748

No rounding needed? It's a whole number

8748 = 4 \cdot 3^{n-1} \\
\\ \displaystyle\frac{8748}{4} = 3^{n-1} \\ \\ 
2187 = 3^{n-1} \\ \\
\log_3(2187) = n - 1 \implies n = \log_3(2187)  + 1 \implies \\ \\ \\
n = \frac{\ln 2187}{\ln 3} + 1 = 7 + 1 = 8

\displaystyle S_n=\frac{a(1-r^n)}{1-r} \implies S_8 =\displaystyle \frac{4(1 - 3^8)}{1 - 3} \\ \\ \\
= 13120

Sum is 13120
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The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the co
statuscvo [17]

Answer:

Step-by-step explanation:

Hello!

The objective is to test if rising the process temperature reduces the thickness of the plastic film that coats a substrate material To do so, two samples of substrates are coated at different temperatures:

Sample 1

X₁: Thickness of the plastic film after the substrate is coated at 125F

n₁=11

X[bar]₁= 101.28

S₁= 5.08

Sample 2

X₂: Thickness of the plastic film after the substrate is coated at 150F

n₂= 13

X[bar]₂= 101.70

S₂= 20.15

Does the data support this claim? Use the P-value approach and assume that the two population standard deviations are not equal.

Now if the higher the heat, the thinner the thickness of the plastic coating, then the average thickness of the coating done at 150F should be less than the average thickness of the coating done at 125F, symbolically: μ₂ < μ₁

Then the hypotheses are:

H₀: μ₂ ≥ μ₁

H₁: μ₂ < μ₁

α:0.05 (there is no α level stated so I've chosen the most common one)

Assuming that both variables have a normal distribution since the population standard deviations are not equal, the statistic to use is the Welch's t-test:

t= \frac{(X[bar]_2-X[bar]_1)-(Mu_1-Mu_2)}{\sqrt{\frac{S^2_1}{n_1} +\frac{S^2_2}{n_2} } } ~~t_w

t_{H_0}= \frac{(101.7-101.28)-0}{\sqrt{\frac{(20.15)^2}{13} +\frac{(5.08)^2}{11} } } = 0.072

This test is one-tailed to the left, meaning that you'll reject the null hypothesis at small values of t. The p-value is also one-tailed and has the same direction as the test. To calculate it you have to first calculate the degrees of freedom of the Welch's t:

Df_w= \frac{(\frac{S^2_1}{n_1} +\frac{S^2_2}{n_2} )^2}{\frac{(\frac{S^2_1}{n_1})^2 }{n_1-1}+\frac{(\frac{S^2_2}{n_2} )^2}{n_2-1}  }

Df_w= \frac{(\frac{5.08^2}{11} +\frac{20.15^2}{13} )^2}{\frac{(\frac{5.08^2}{11})^2 }{10} +\frac{(\frac{20.15^2}{13} )^2}{12} } = 13.78

The distribution is a Student's t with 13 degrees of freedom, then you can calculate the p-value as:

P(t₁₃≤0.072)= 0.4718

Using the p-value approach, the decision rule is:

If the p-value ≤ α, the decision is to reject the null hypothesis.

If the p-value > α, the decision is to not reject the null hypothesis.

The p-value is greater than the significance level, so the decision is to nor reject the null hypothesis.

Using a significance level of 5%, there is no significant evidence to support the claim that the average thickness pf the plastic coat processed at 150F is less than the average thickness pf the plastic coat processed at 125F.

I hope you have a SUPER day!

4 0
3 years ago
What are the answers to these angle questions?​
mafiozo [28]

Step-by-step explanation:

Firstly, we have to find m∠J.

Since all the angles of a Δ equal 180°, angles J, L, and K should have a sum of 180°.

So,

m∠J + m∠L + m∠K = 180°

The diagram shows us that ∠L = 49° and ∠K = 90°, so we plug in those numbers in the equation.

m∠J + 49° + 90° = 180°

Then we simplify

m∠J + 139° = 180°

Subtract 139° to both sides

∠J = 41

Now the other angles.

Since ΔJKL ~ ΔRST, then ∠J ≅ ∠R, ∠K ≅ ∠S, and ∠L ≅ ∠T

Meaning, m∠J = m∠R, m∠K = m∠S, and m∠L = m∠T

Since we know m∠J = 41°, m∠K = 90°, and m∠L = 49° we could plug those in so...

41° = m∠R , 90° = m∠S , and 49° = m∠T

4 0
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Chapter : EXPONENTS AND POWERS<br> Can anyone give me answer for this please
Aneli [31]

The value of given expression is: 3

<h3>What is exponents and powers?</h3>

Exponent refers to the number of times a number is used in a multiplication. Power can be defined as a number being multiplied by itself a specific number of times.

9*(15)³/45 * 5^{-2}/9

= 3²* 3³*5³/5² * 3² * 3² * 5

= 3^{2+3-4} * 5^{3-3}

= 3*1

=3

Learn more about exponents and power here:

brainly.com/question/15722035

#SPJ1

5 0
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Richard walks every day for exercise at a rate of 1 kilometer every 12 min. At this rate ,how many meters can Richard walk in a
garik1379 [7]
60 minutes in an hour
60÷12=5
1,000 meter in 1 km
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5×1000=5000
therefore, 5000 meters
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36 of them did not have an umbrella because out of 11 people, 4 did not bring an umbrella.
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