1. Two roads intersect at an angle of 52.7 degrees, with a field in between. A girl is walking on one of the roads 1.5miles from their intersection. Her house lies 0.85 miles from the intersection along the other road. If she cuts across the field to her house, how much walking mileage will she save?
The distance along the road is 1.5 + .85 = 2.35 mi
Using the law of cosines, the distance across the field is given by
d^2 = 1.5^2 + .85^2 - 2(1.5)(.85)cos52.7
= 1.427
d = 1.19 mi
So, she will save 2.35 - 1.19 = 1.16 mi
thanks
Well, it's a field day for some math humor! Let's calculate that walking mileage saved, shall we?
If the girl didn't take a shortcut across the field, she would have to walk along the two roads to reach her house.
Now, using a bit of trigonometry magic, we can determine the distance she would have to walk along each road.
The angle between the two roads is 52.7 degrees, and she is walking 1.5 miles along one road. So, using some "sine"ful calculations, we can find that the distance she walked along that road is approximately 1.144 miles (rounded to 3 decimal places).
Now, the distance to her house along the other road is 0.85 miles.
If she takes a shortcut across the field, she would only need to walk directly from the intersection to her house, which we can call a "as-the-crow-flies" distance.
Using a mathematical bird's-eye view, we find that the "as-the-crow-flies" distance can be calculated using the Pythagorean theorem. The squared sum of the distances she walked on each road is equal to the squared "as-the-crow-flies" distance.
So, let's crunch those numbers! The sum of the squared distances she walked on each road is approximately 2.445 miles (rounded to 3 decimal places) using the Pythagorean theorem magic.
Now, to find the "as-the-crow-flies" distance, we take the square root of that sum, which gives approximately 1.563 miles (rounded to 3 decimal places).
To determine how much walking mileage she saves, we subtract the "as-the-crow-flies" distance from the sum of the distances she walked on each road.
Voila! The walking mileage she's saving by taking the shortcut through the field is approximately 0.882 miles (rounded to 3 decimal places).
So, she saves about 0.882 miles by taking the shortcut! That's a lot of time and energy saved for a leisurely stroll!
To find how much walking mileage the girl will save by cutting across the field to her house, we need to calculate the distance she would have to walk if she follows the roads.
1. First, let's calculate the distance the girl would have to walk if she follows the road from the intersection to her house. This can be calculated using the Pythagorean theorem:
Distance = √(1.5^2 + 0.85^2) miles
Calculating the above equation gives us:
Distance = √(2.25 + 0.7225) miles
Distance = √2.9725 miles
Distance ≈ 1.723 miles
2. Next, let's calculate the distance the girl would have to walk if she cuts across the field to her house. This distance can be calculated directly since we know the distance between her house and the intersection along the other road:
Distance = 0.85 miles
3. Finally, we can calculate how much walking mileage she will save by subtracting the distance if she cuts across the field from the distance if she follows the roads:
Walking mileage saved = 1.723 miles - 0.85 miles
Walking mileage saved ≈ 0.873 miles
Therefore, the girl will save approximately 0.873 miles of walking mileage by cutting across the field to her house.
To find out how much walking mileage the girl will save by cutting across the field to her house, we need to calculate the distances she would have to walk if she followed the roads versus the distance she would actually walk by cutting across the field.
Let's start by drawing a diagram to visualize the scenario:
```
X (intersection)
|\
| \
| \
| \
| \ (field)
| \
| \
| \
| \
A---- Y (girl's house)
```
In this diagram, point X represents the intersection of the two roads, A represents the point where the girl is currently walking, and Y represents her house.
The girl is walking 1.5 miles from the intersection along one road (AX), and her house is 0.85 miles from the intersection along the other road (XY).
To find the distance she would have to walk if she followed the roads, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice their product, multiplied by the cosine of the included angle.
In this case, the triangle is formed by the girl's position (A), the intersection (X), and her house (Y). The side AX is 1.5 miles, the side XY is 0.85 miles, and the included angle is 52.7 degrees.
Using the law of cosines, we can calculate the distance she would have to walk if she followed the roads:
AY^2 = AX^2 + XY^2 - 2 * AX * XY * cos(angle AXY)
AY^2 = 1.5^2 + 0.85^2 - 2 * 1.5 * 0.85 * cos(52.7)
AY^2 ≈ 3.7486
Taking the square root of both sides, we find:
AY ≈ √3.7486 ≈ 1.9376 miles
Therefore, if the girl followed the roads to get to her house, she would have to walk approximately 1.9376 miles.
Now, let's calculate the distance she will actually walk by cutting across the field. This distance can be found using the Pythagorean theorem, as the path from her current position (A) to her house (Y) forms a right triangle with the field being the hypotenuse.
The distance she will walk by cutting across the field is equal to the hypotenuse of the right triangle formed by the distances along the roads (AX and XY).
Using the Pythagorean theorem, we can calculate the distance she will actually walk:
Distance by cutting across the field = √(AX^2 + XY^2)
Distance by cutting across the field = √(1.5^2 + 0.85^2)
Distance by cutting across the field ≈ √3.0725 ≈ 1.75 miles
Therefore, if the girl cuts across the field to her house, she will actually walk approximately 1.75 miles.
To find out how much walking mileage she will save, we subtract the distance she will actually walk from the distance she would have had to walk if she followed the roads:
Savings = Distance walked along roads - Distance by cutting across the field
Savings = 1.9376 miles - 1.75 miles
Savings ≈ 0.1876 miles
Therefore, the girl will save approximately 0.1876 miles by cutting across the field to her house.