For positive acute angles
1 answer:
9514 1404 393
Answer:
77/85
Step-by-step explanation:
<u>Given</u>:
cos(A) = 4/5
sin(B) = 15/17
<u>Find</u>:
cos(A-B)
<u>Solution</u>:
From your knowledge of Pythagorean triples, you know that one of them is (3, 4, 5) and another is (8, 15, 17). Using your imagination, or by drawing the triangles, you can determine the needed trig functions to be ...
sin(A) = 3/5
cos(B) = 8/17
__
Alternatively, you can use the Pythagorean identity to find ...
sin(A) = √(1 -cos²(A)) = √(1 -(4/5)²) = √(9/25) = 3/5
cos(B) = √(1 -sin²(B)) = √(1 -(15/17)²) = √(64/289) = 8/17
__
Then application of the angle difference formula is straightforward.
cos(A -B) = cos(A)cos(B) +sin(A)sin(B)
cos(A -B) = (4/5)(8/17) +(3/5)(15/17) = (32 +45)/85
cos(A -B) = 77/85
_____
Some graphing calculators can express the result ...
cos(arccos(4/5) -arcsin(15/17)) ≈ 0.90588235294
... as the fraction 77/85
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Answer:
a) 79.67% probability that a randomly selected bottle has at least 355 ml
b) 99.29% probability that the bottles of a randomly selected 6-pack of beer have an average of at least 355 ml.
c) 88.88% probability that the total amount of beer in the 6-pack is less than 2131.5 ml
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question:
a. What is the probability that a randomly selected bottle has at least 355 ml?
This is 1 subtracted by the pvalue of Z when X = 355. So
has a pvalue of 0.2033
1 - 0.2033 = 0.7967
79.67% probability that a randomly selected bottle has at least 355 ml.
b. What is the probability that the bottles of a randomly selected 6-pack of beer have an average of at least 355 ml?
Now
By the Central Limit Theorem
has a pvalue of 0.0071
1 - 0.0071 = 0.9929
99.29% probability that the bottles of a randomly selected 6-pack of beer have an average of at least 355 ml.
c. What is the probability that the total amount of beer in the 6-pack is less than 2131.5 ml?
2131.5/6 = 355.25
This is the pvalue of Z when X = 355.25.
has a pvalue of 0.1112
1 - 0.1112 = 0.8888
88.88% probability that the total amount of beer in the 6-pack is less than 2131.5 ml