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A ladder 12ft long leans against a building and makes a 32 degree angle with level ground. How far up the building does the ladder reach? What type of angle is this? Draw a picture and find the angle

A ladder 12ft12\text{ft} long leans against a building and makes a 3232 degree angle with level ground. How far up the building does the ladder reach? What type of angle is this? Draw a picture and find the angle

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Q. A ladder 12ft12\text{ft} long leans against a building and makes a 3232 degree angle with level ground. How far up the building does the ladder reach? What type of angle is this? Draw a picture and find the angle
  1. Trigonometry Formula: To solve this problem, we will use trigonometry. Specifically, we will use the cosine function, which relates the adjacent side of a right-angled triangle (the distance up the building) to the hypotenuse (the length of the ladder). The formula is:\newlinecos⁡(angle)=adjacent sidehypotenuse\cos(\text{angle}) = \frac{\text{adjacent side}}{\text{hypotenuse}}
  2. Convert Angle to Radians: First, we need to convert the angle from degrees to radians because some calculators require it. However, in this case, most calculators have a degree mode, so we can use the angle in degrees directly. We will assume the calculator is set to degree mode.
  3. Plug in Values: Now, we can plug in the values we know into the cosine formula. We have the angle 32∘32^\circ and the hypotenuse 12 ft12\,\text{ft}. We need to find the adjacent side, which is the distance up the building the ladder reaches.cos⁡(32∘)=adjacent side12 ft\cos(32^\circ) = \frac{\text{adjacent side}}{12\,\text{ft}}
  4. Calculate Adjacent Side: To find the adjacent side, we multiply both sides of the equation by the hypotenuse 12ft12ft: extadjacentside=cos⁡(32∘)×12ft ext{adjacent side} = \cos(32^\circ) \times 12ft
  5. Calculate Adjacent Side: Using a calculator, we find the cosine of 3232 degrees and then multiply by 12 ft12\,\text{ft}:\newlineadjacent side=cos⁡(32 degrees)×12 ft≈0.8480×12 ft≈10.176 ft\text{adjacent side} = \cos(32 \text{ degrees}) \times 12\,\text{ft} \approx 0.8480 \times 12\,\text{ft} \approx 10.176\,\text{ft}
  6. Type of Angle: The type of angle made by the ladder with the ground is an acute angle because it is less than 9090 degrees.
  7. Draw Triangle: To draw a picture, sketch a right-angled triangle with the ladder as the hypotenuse, the ground as the base, and the height as the side of the building. The angle between the ladder and the ground is 3232 degrees.
  8. Find Remaining Angle: To find the angle between the ladder and the building, we can use the fact that the sum of angles in a triangle is 180180 degrees. Since one angle is 9090 degrees (right angle) and the other is 3232 degrees, the remaining angle is:\newline180180 degrees −- 9090 degrees −- 3232 degrees == 5858 degrees

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