A cow is tethered by a 110-ft rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze. (Let
a = 10 ft, b = 60 ft, c = 110 ft, d = 70 ft, and e = 60 ft.
Round your answer to the nearest whole number.)
Well, I must say, this cow seems to have quite the space to roam! Let's see what we can do with those numbers.
Now, the area that the cow can graze is essentially the area inside the L-shaped building that falls within the reach of the 110-ft rope. So, the first thing we need to do is figure out the shape of the grazing area.
From the given dimensions, we can see that one side of the L-shaped building is 60 ft long (let's call it side "b"), while the other side is 70 ft long (let's call it side "d"). These two sides meet at a right angle, forming a nice little corner for the cow to chill.
Now, let's consider the scenario when the cow is standing at the inside corner of the building, tethered by the 110-ft rope. Picture it in your mind, the cow going in circles, chewing grass, and maybe telling some funny jokes to keep itself entertained.
Since the rope is 110 ft long (let's call it side "c"), we know that the grazing area can be thought of as a circle centered at the inside corner, with a radius of 110 ft.
To calculate the area of this circle, we use the good ol' formula: Area = π * r^2, where r is the radius. Plugging in our values, we get:
Area = π * (110 ft)^2
Now, rounding our answer to the nearest whole number, we find that the cow can graze approximately 38,026 square feet.
So, there you have it! The cow has a grazing area of about 38,026 square feet to munch on. Just imagine all the delicious grass it can get its hooves on! Happy grazing, cow!
To find the area that the cow can graze, we need to determine the shape of the grazing area. Let's break it down step by step:
Step 1: Identify the shape of the grazing area.
Looking at the figure and information provided, we can see that the grazing area is a combination of a rectangle and a quarter circle.
Step 2: Find the dimensions of the rectangle.
The dimensions of the rectangle will be equal to the dimensions of the L-shaped building. Given that a = 10 ft and b = 60 ft, the dimensions of the rectangle are 10 ft by 60 ft.
Step 3: Find the dimensions of the quarter circle.
The quarter circle is formed by the circular arc with radius 110 ft and the straight side of length 60 ft. To find the dimensions of the quarter circle, we need to determine the length of the circular arc and the radius.
Step 4: Find the length of the circular arc.
The length of the circular arc can be calculated using the formula: length = (angle / 360) * 2 * π * radius. In this case, we know the angle is 90 degrees (a quarter circle) and the radius is 110 ft. Plugging in these values, we get:
length = (90 / 360) * 2 * π * 110 = (1/4) * 2 * π * 110 = 0.25 * 2 * 3.14 * 110 = 173.5 ft (rounded to the nearest whole number).
Step 5: Find the radius of the quarter circle.
The radius of the quarter circle is the length of the side of the L-shaped building, which is 70 ft (given as d).
Step 6: Calculate the area of the rectangle.
The area of the rectangle can be found using the formula: area = length * width. Plugging in the values, we get:
area = 10 ft * 60 ft = 600 ft^2.
Step 7: Calculate the area of the quarter circle.
The area of a quarter circle can be found using the formula: area = (π * r^2) / 4, where r is the radius. Plugging in the values, we get:
area = (3.14 * 70 ft^2) / 4 = 154 ft^2 (rounded to the nearest whole number).
Step 8: Find the total area of the grazing area.
To find the total area, we need to add the area of the rectangle and the area of the quarter circle. Therefore, the total area the cow can graze is:
total area = area of rectangle + area of quarter circle = 600 ft^2 + 154 ft^2 = 754 ft^2 (rounded to the nearest whole number).
Therefore, the cow can graze an area of approximately 754 square feet.
To find the area that the cow can graze, we need to determine the shape of the grazing area. From the given information, we can see that the grazing area is formed by two rectangles and two triangles.
First, let's start by calculating the area of the rectangles:
Rectangle A (with length b and a width of 110 ft):
Area A = b * 110 = 60 * 110 = 6,600 square ft
Rectangle B (with length c and a width of d):
Area B = c * d = 110 * 70 = 7,700 square ft
Now, let's calculate the area of the triangles:
Triangle C:
The base of this triangle is side a, and the height is 60 ft.
Area C = (1/2) * (a * 60) = (1/2) * (10 * 60) = 300 square ft
Triangle D:
The base of this triangle is side e, and the height is side d.
Area D = (1/2) * (e * d) = (1/2) * (60 * 70) = 2,100 square ft
Now, we can find the total area by summing up the areas of the rectangles and triangles:
Total Area = Area A + Area B + Area C + Area D
= 6,600 + 7,700 + 300 + 2,100
= 16,700 square ft
Therefore, the cow can graze an area of approximately 16,700 square feet.