Mathematics College

Louie is trying to find a rectangular canvas for his art project. Its width must measure 20 inches and form a 35° angle with the diagonal of the canvas. What is the height of the canvas? Round your answer to the nearest tenth.

11.5 inches
14.0 inches
16.4 inches
28.6 inches

Answers

Answer 1

The height of the canvas is approximately 11.5 inches.

To find the height of the canvas, we can use trigonometry and the given angle. Let's denote the height as 'h' inches.

We know that the width of the canvas is 20 inches, and it forms a 35° angle with the diagonal. Since a rectangle has 90° angles, the diagonal forms a right triangle with the width and height of the canvas.

The diagonal can be found using the Pythagorean theorem: diagonal^2 = width^2 + height^2.

In this case, diagonal^2 = 20^2 + h^2.

Since the angle between the width and the diagonal is given as 35°, we can use the sine function: sin(35°) = opposite/hypotenuse = h/diagonal.

Rearranging the equation, we have h = diagonal * sin(35°).

Substituting the value of diagonal^2 from the Pythagorean theorem, we get h = sqrt(20^2 + h^2) * sin(35°).

Solving this equation, we find that h ≈ 11.5 inches.

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Related Questions

Find the missing measurement in the figure below (angles 1-6)

Answers

Using angle rules, the values of angles 1 to 6 are 64, 53, 116, 89, 32 and 44 respectively.

Angle 1

Angle 1 + 69 + 47 = 180 (sum of angles on a straight line)

Angle 1 = 180 - (69+47)

Angle 1 = 64°

Angle 2

Angle 2 + 64 + 63 = 180 (sum of angles in a triangle)

Angle 2 = 180 - (64+63)

Angle 2 = 53°

Angle 3

Angle 1 + Angle 3 = 180 (sum of angles in a triangle)

Angle 3 = 180 - 64

Angle 3 = 116°

Angle 6

Angle 6 + 136 = 180 (sum of angles on a straight line)

Angle 6 = 180 - 136

Angle 6 = 44°

Angle 5

Angle 5 + Angle 3 + 32 = 180 (sum of angles in a triangle)

Angle 5 + 116 + 32 = 180

Angle 5 = 180 - (116 + 32)

Angle 5 = 32°

Angle 4

47 + Angle 6 + Angle 4 = 180

47 + 44 + Angle 4 = 180

Angle 4 = 180 - 91

Angle 4 = 89°

Therefore, the values of angles 1 to 6 are 64, 53, 116, 89, 32 and 44 respectively.

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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.

Answers

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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The figure below shows a triangular piece of cloth:
7 in.
What is the length of the portion BC of the cloth?
07 cos 33°
sin 33
07 sin 33°
B
O cos 33

Answers

The length of portion BC of the cloth is approximately 5.8709 inches.

To find the length of portion BC of the cloth, we need to use trigonometric functions.

In this case, we can use the cosine function.

Given that the adjacent side to angle B is labeled BC and the hypotenuse is labeled 7 in, we can apply the cosine function, which is defined as the adjacent side divided by the hypotenuse:

cos(angle) = adjacent / hypotenuse

In this scenario, the angle we are considering is 33 degrees.

Therefore, we have:

cos(33°) = BC / 7 in

To isolate BC, we can rearrange the equation:

BC = 7 in [tex]\times[/tex] cos(33°)

Calculating this expression, we find:

BC ≈ 7 in [tex]\times[/tex] 0.8387 (rounded to four decimal places)

BC ≈ 5.8709 in.

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Find the area of a composite figure.

Answers

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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