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leva [86]
2 years ago
11

Show that the area of triangle DEF is 11 cm2.​

Mathematics
1 answer:
dedylja [7]2 years ago
5 0

9514 1404 393

Explanation:

The length CD can be found from the Pythagorean theorem.

  CF² + CD² = DF²

  2² + CD² = (2√10)²

  CD² = 40 -4 = 36

  CD = √36 = 6

From here, any of several methods could be used to find the area of ΔDEF.

One of them is to subtract the areas of the triangles that are not ΔDEF from the area of the rectangle ABCD.

  area ABCD = bh = (6 cm)(5 cm) = 30 cm²

  area ΔADE = 1/2bh = 1/2(4 cm)(5 cm) = 10 cm²

  area ΔBEF = 1/2bh = 1/2(2 cm)(3 cm) = 3 cm²

  area ΔCDF = 1/2bh = 1/2(6 cm)(2 cm) = 6 cm²

Then the area of interest is ...

  area ΔDEF = area ABCD -area ΔADE -area ΔBEF -area ΔCDF

  area ΔDEF = (30 -10 -3 -6) cm²

  area ΔDEF = 11 cm²

__

<em>Alternate solution</em>

If we define A as the origin of a coordinate system, then the coordinates of the vertices of ΔDEF are ...

  D(0, 5), E(4, 0), F(6, 3)

The area can be computed as half the absolute value of the sum of the determinants of 2×2 matrices consisting of adjacent points.

  A=\dfrac{1}{2}\left|\left|\begin{array}{cc}0&5\\4&0\end{array}\right|+\left|\begin{array}{cc}4&0\\6&3\end{array}\right|+\left|\begin{array}{cc}6&3\\0&5\end{array}\right| \right|\\\\=\dfrac{1}{2}|(0-20)+(12-0)+(30-0)|=\dfrac{1}{2}(22) = \boxed{11}

_____

<em>Additional comment</em>

The solution using coordinates is effectively equivalent to the computation of the area of trapezoid ABFD, less the areas of triangles EBF and AED.

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Answer:

19)  ( \frac{-2}{5})^2 = \frac{4}{25}

20) -9^{2} = -81

21)  2^{4} = 16

Step-by-step explanation:

The only wrong ones are

19)  ( \frac{-2}{5})^2 = \frac{4}{25}

20) -9^{2} = -81 ... negative is not inside a parenthesis that's why it is still negative

21) you can add another way is 2^{4} = 16

Everything else looks good.  Great job!

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Step-by-step explanation:

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4 0
3 years ago
A pathologist knows that 14.9% of all deaths are attributable to myocardial infarctions (a type of heart disease). (Either you h
garik1379 [7]

Answer:

a) The mean number is 745 and the standard deviation is 25.18.

b) 896 deaths is a significantly high number, which means that there is cause for concern.

Step-by-step explanation:

For each death, there are only two possible outcomes. Either it is attributable to myocardial infarctions, or it is not. The probability of a death being attributable to myocardial infarctions is independent of other deaths. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

\sqrt{V(X)} = \sqrt{np(1-p)}

A value X is significantly high is:

X > E(X) + 2\sqrt{V(X)}

In this problem:

p = 0.149, n = 5000

a. Find the mean and standard deviation for the number of such deaths that will occur in typical region with 5000 deaths.

E(X) = np = 5000*0.149 = 745

\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{5000*0.149*0.851} = 25.18

The mean number is 745 and the standard deviation is 25.18.

b. In one region, 5000 death certificates are examined, and it is found that 896 deaths were attributable to myocardial infarction. Is there cause for concern? Why or why not?

Is 896 a significantly high number?

E(X) + 2\sqrt{V(X)} = 745 + 2*25.18 = 795.36

895 > 795.36

896 deaths is a significantly high number, which means that there is cause for concern.

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