You set up a makeshift greenhouse by leaning a square pane of glass against a building. The glass is 4.5 ft long and it makes a 30 degree angle with the ground. How much horizontal distance between the building and the glass is there to grow plants? Round to the nearest inch.
To find the horizontal distance between the building and the glass, we can use trigonometry. Let's call this distance "x."
First, let's visualize the problem. The square pane of glass is leaning against the building, forming a right triangle with the ground. The 30-degree angle is the angle between the ground and the glass.
Next, we need to identify the sides of the right triangle. The vertical side of the triangle is the height of the glass against the ground, which is 4.5 feet. The horizontal side of the triangle is the distance between the building and the glass (x), and the hypotenuse represents the leaning glass.
The trigonometric function we need to use is the sine function (sin), which relates the length of the sides in a right triangle.
In this case, sin(30 degrees) = opposite/hypotenuse, which means sin(30 degrees) = 4.5 ft / hypotenuse.
Applying the trigonometric function, we can solve for the hypotenuse: hypotenuse = 4.5 ft / sin(30 degrees).
Calculating sin(30 degrees) gives us approximately 0.5, so we have: hypotenuse = 4.5 ft / 0.5.
Dividing 4.5 ft by 0.5 gives us a hypotenuse length of 9 ft.
Finally, to find the horizontal distance (x) between the building and the glass, we use the cosine function (cos):
cos(30 degrees) = adjacent/hypotenuse, which means cos(30 degrees) = x / 9 ft.
Rearranging the equation, we can solve for x: x = 9 ft * cos(30 degrees).
Calculating cos(30 degrees) gives us approximately 0.866, so we have: x ≈ 9 ft * 0.866.
Multiplying 9 ft by 0.866 gives us an approximate horizontal distance (x) of 7.794 ft.
To round the result to the nearest inch, we can multiply the decimal part (0.794) by 12 inches, since there are 12 inches in a foot. This gives us approximately 9.528 inches.
Therefore, the horizontal distance between the building and the glass, rounded to the nearest inch, is approximately 7 feet and 9.5 inches.
To determine the horizontal distance between the building and the glass, we can use trigonometry. Specifically, we can use the sine function since we have the length of the glass and the angle it makes with the ground.
The formula for the horizontal distance (x) is given by:
x = length of glass * sin(angle)
Given that the length of the glass is 4.5 ft and the angle is 30 degrees, we can plug these values into the formula:
x = 4.5 ft * sin(30°)
To find the value of sin(30°), we can consult a trigonometric table or use a calculator. For simplicity, we will approximate it as 0.5.
x = 4.5 ft * 0.5
x = 2.25 ft
The horizontal distance between the building and the glass is approximately 2.25 feet.
make a sketch, label the base distance as x
then cos 30° = x/4.5
x = 4.5cos30° = 3.897 inches or 4 inches to the nearest inch