To find the measures of angles ABD and DBC in the right triangle ABC, we considered two cases: when point D lies on segment AB and when point D lies on segment BC. In both cases, the measures of the angles were determined based on the given ratio of 3:2.
In the given problem, we have a right triangle ABC, and we need to find the measures of angle ABD (mABD) and angle DBC (mDBC) based on the ratio of the measures of angle A and angle B.
Let's consider the different cases for the location of point D:
Case 1: Point D lies on segment AB.
In this case, angle ABD and angle DBC will be acute angles. Let's assume that angle ABD has a measure of 3x and angle DBC has a measure of 2x. Since angle ABD and angle DBC are acute angles, the sum of their measures should be less than 90 degrees.
Therefore, we have the inequality: 3x + 2x < 90. Solving this inequality, we get 5x < 90, which gives x < 18. So, the measure of angle ABD (mABD) will be 3x, and the measure of angle DBC (mDBC) will be 2x.
Case 2: Point D lies on segment BC.
In this case, angle ABD and angle DBC will be obtuse angles. Let's assume that angle ABD has a measure of 3x and angle DBC has a measure of 2x. Since angle ABD and angle DBC are obtuse angles, the sum of their measures should be greater than 90 degrees.
Therefore, we have the inequality: 3x + 2x > 90. Solving this inequality, we get 5x > 90, which gives x > 18. So, the measure of angle ABD (mABD) will be 3x, and the measure of angle DBC (mDBC) will be 2x.
By investigating these two cases, we can find the measures of angle ABD (mABD) and angle DBC (mDBC) based on the given ratio of 3:2. The specific values of mABD and mDBC will depend on the exact location of point D within the triangle ABC.
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In a controlled laboratory experiment, scientists at the University of Minnesota discovered that
25% of a certain strain of rats subjected to a 20% coffee
bean diet and then force-fed a powerful cancer-causing
chemical later developed cancerous tumors. Would we
have reason to believe that the proportion of rats developing tumors when subjected to this diet has increased
if the experiment were repeated and 16 of 48 rats developed tumors? Use a 0.05 level of significance.
Yes, we would have reason to believe that the proportion of rats developing tumors when subjected to this diet has increased if the experiment were repeated and 16 of 48 rats developed tumors.
To determine whether there is an increase in the proportion of rats developing tumors when subjected to a coffee bean diet, we can conduct a hypothesis test using the 0.05 level of significance.
1. State the hypotheses:
- Null hypothesis (H0): The proportion of rats developing tumors remains the same.
- Alternative hypothesis (Ha): The proportion of rats developing tumors has increased.
2. Identify the test statistic:
We will use a z-test to compare the observed proportion of rats developing tumors with the expected proportion.
3. Set the significance level:
The significance level (α) is given as 0.05.
4. Collect data:
In the original experiment, 25% of rats developed tumors. In the repeated experiment, 16 out of 48 rats developed tumors.
5. Compute the test statistic:
The test statistic formula for comparing proportions is:
z = (p - P) / sqrt(P(1-P)/n)
where p is the observed proportion, P is the hypothesized proportion, and n is the sample size.
Using the observed proportion (16/48 = 0.333), the hypothesized proportion (0.25), and the sample size (48), we can calculate the test statistic.
6. Determine the critical value:
Since we are using a 0.05 level of significance and conducting a one-tailed test (Ha: >), we can find the critical value from the standard normal distribution table. The critical value for a 0.05 significance level is 1.645.
7. Make a decision:
If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the proportion of rats developing tumors has increased.
8. Calculate the test statistic:
Plugging in the values into the formula, we calculate the test statistic:
z = (0.333 - 0.25) / sqrt(0.25 * 0.75 / 48) = 1.404
9. Compare the test statistic and critical value:
The test statistic (1.404) is less than the critical value (1.645).
10. Make a decision:
Since the test statistic is not greater than the critical value, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the proportion of rats developing tumors has increased when subjected to this diet.
In summary, based on the given data and conducting a hypothesis test, we do not have reason to believe that the proportion of rats developing tumors has increased if the experiment were repeated and 16 of 48 rats developed tumors.
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Find the area of a composite figure.
The area of the composite figure is 800m²
What is area of a figure?The area of a figure is the number of unit squares that cover the surface of a closed figure.
Composite geometric figures are made from two or more geometric figures.
The figure consist of a rectangle , a semi circle and a triangle.
Area of the semicircle = 1/2 πr²
= 1/2 × 3.14 × 10²
= 314/2 = 157 m²
Area of the rectangle = l × w
= 25 × 20
= 500m²
area of the triangle = 1/2bh
= 1/2 × 10 × 25
= 25 × 5
= 125 m²
Therefore the area of the composite figure
= 125 + 500 + 175
= 800m²
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