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Matt is standing on top of a cliff 305 feet above a lake. The measurement of the angle of depression to a boat on the lake is 42
1 answer:
Answer:
≈ 410.4427 feet (this is rounded -- it's an infinite decimal)
Step-by-step explanation:
If you draw it out, you get a right triangle. If Matt is standing 305 feet above a lake, that is the altitude of the triangle. The angle of depression means, for example, the angle looking down from something at something else, in this case, from the cliff to the boat. The 42° is the acute angle between the altitude and the hypotenuse, the one at the top of the triangle. You want to find how far away the boat is from Matt directly, so that would be the hypotenuse, just connecting a straight line from him to the boat.
To answer these problems there are three tools you use to find the answer: Sine, Cosine, and Tangent. Let's see which one would best help us in this situation.
The definition of sine is the side opposite the acute angle over the hypotenuse.
The definition of cosine is the side adjacent to the acute angle over the hypotenuse.
The definition of tangent is the side opposite the acute angle over the side adjacent to it.
In this case, you know you have the side adjacent to the acute angle, over the hypotenuse. Because you have the acute angle (42°) and the side adjacent to it (305) and the hypotenuse (which is what you're trying to find the value of. Which definition would fit this figure? Cosine, because it's side adjacent over hypotenuse.
Once you have that straightened out, you know that you need to find the cosine of 42°, which, if you look in a sine, cosine, and tangent chart (or type 42 into your calculator and hit cos) is 0.7431. Now you can write the equation .
Working with that equation, you need to get x by itself to figure out its value. How you do that is by multiplying both sides by x. This would look like . When you multiply 305 by x, you are multiplying by the number you were just dividing by so that answer is 305. I'll give you an example with different numbers: 6/2 then 6*2 is just 6.
What your equation looks like now is . How do you find x? Well it's the same as doing this:
. Then you would divide 8 by 2 to get x, which equals 4. So, going back to our task at hand,
. That's your answer, which is how far Matt is from the boat. Depending on which digit they want you to round the answer to you could punch in 305/0.7431 into a calculator and see many more digits and a much longer number. But don't forget to include units in your answer -- feet.
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Answer:
The fraction of smoothies sold were either banana or strawberry is
Step-by-step explanation:
Given:
Of the smoothies sold yesterday at Marlins smoothie shop, 5/6 were banana and another 1/12 were strawberry.
Now, to find the fraction of the smoothies sold were either banana or strawberry.
Fraction of smoothie were of banana =
Fraction of smoothie were of strawberry =
Now, to get the fraction smoothies sold were either banana or strawberry by adding both the fractions of banana and strawberry:
Therefore, the fraction of smoothies sold were either banana or strawberry is
Answer:
a) The 99% confidence interval is given by (0.198;0.242).
b) Based on the p value obtained and using the significance level assumed we have
so we can conclude that we fail to reject the null hypothesis, and we can said that at 1% of significance the proportion of people who are rated with Excellent/Good economy conditions not differs from 0.24. The interval also confirms the conclusion since 0.24 it's inside of the interval calculated.
c)
Step-by-step explanation:
<em>Data given and notation </em>
n=2362 represent the random sample taken
X represent the people who says that they would watch one of the television shows.
estimated proportion of people rated as Excellent/Good economic conditions.
is the value that we want to test
represent the significance level
z would represent the statistic (variable of interest)
represent the p value (variable of interest) <em>
</em>
<em>Concepts and formulas to use </em>
We need to conduct a hypothesis in order to test the claim that 24% of people are rated with good economic conditions:
Null hypothesis:
Alternative hypothesis:
When we conduct a proportion test we need to use the z statistic, and the is given by:
(1)
The One-Sample Proportion Test is used to assess whether a population proportion is significantly different from a hypothesized value
.
Part a: Test the hypothesis
<em>Check for the assumptions that he sample must satisfy in order to apply the test </em>
a)The random sample needs to be representative: On this case the problem no mention about it but we can assume it.
b) The sample needs to be large enough
np = 2362x0.22=519.64>10 and n(1-p)=2364*(1-0.22)=1843.92>10
Condition satisfied.
<em>Calculate the statistic</em>
Since we have all the info requires we can replace in formula (1) like this:
The confidence interval would be given by:
The critical value using and
would be
. Replacing the values given we have:
So the 99% confidence interval is given by (0.198;0.242).
Part b
<em>Statistical decision </em>
P value method or p value approach . "This method consists on determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided is . The next step would be calculate the p value for this test.
Since is a bilateral test the p value would be:
So based on the p value obtained and using the significance level assumed we have
so we can conclude that we fail to reject the null hypothesis, and we can said that at 1% of significance the proportion of people who are rated with Excellent/Good economy conditions not differs from 0.24. The interval also confirms the conclusion since 0.24 it's inside of the interval calculated.
Part c
The confidence level assumed was 99%, so then the signficance is given by