Find the distance between the points (4,3) and (-1,4) round answer to the nearest tenth
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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:
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:
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!
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Answer:
(2.27, 0.96)
Step-by-step explanation:
The orthocenter is the point of intersection of altitudes of the triangle. The equation for an altitude is the equation of a line through a vertex that is perpendicular to the opposite side.
For example, the line perpendicular to side AC can be found as ...
(∆x, ∆y) = C-A = (3, -1) -(-4, 2) = (7, -3)
∆x(x -h) +∆y(y -k) = 0 . . . . perpendicular through point (h, k)
For side AC, we want the point to be B(4, 5), so the equation is ...
7(x -4) -3(y -5) = 0
7x -3y -13 = 0
Similarly, the altitude to side AB can be written as ...
(∆x, ∆y) = B -A = (4, 5) -(-4, 2) = (8, 3)
8(x -3) +3(y +1) = 0
8x +3y -21 = 0
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By the "cross multiplication method", the solution to these equations is ...
x = (-3(-21) -(3(-13))/(7(3) -8(-3)) = (63+39)/(21+24) = 102/45 = 2 4/15 ≈ 2.27
y = (-13(8) -(-21)(7))/45 = 43/45 ≈ 0.96
The orthocenter is near (2.27, 0.96).
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A graphing application confirms this result.
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<em>Additional information about the cross multiplication method</em>
For the general form equations ...
- ax +by +c = 0
- dx +ey +g = 0
The "cross multiplication method" has you write the array ...
and form the "cross products" in groups of four coefficients:
D = ae -db, X = bg -ec, Y = cd -ga
Then the solution to the set of equations is ...
1/D = x/X = y/Y ⇒ x = X/D, y = Y/D
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<em>Additional note</em>
Videos of this method show you writing the array as bcab/egde and using the final equation x/X=y/Y=1/D. The above gets the same result in a more straightforward manner.
