The diagram shows the plan made by a landscape artist for a homeowner's 16-meter by a 16-meter backyard . The homeowner has requested that rosebushes, fountain, and garden bench be placed so that they are no more that 14 meters apart.
A). Has the landscape artist met the homeowner's requirements? Explain and show work.
B)The homeowner has purchased a sundial to be placed midway between the fountain and the rosebushes. Determine the coordinates indicating where the sundial should be placed.
A) To determine if the landscape artist has met the homeowner's requirements, we need to analyze the given information and see if the distances between the rosebushes, fountain, and garden bench are no more than 14 meters apart.
From the diagram, we can see that the distance between any two adjacent corners of the backyard is 16 meters. Therefore, the diagonal distance between two corners is equal to the hypotenuse of a right-angled triangle with sides measuring 16 meters.
Using the Pythagorean theorem, we can calculate the exact diagonal distance as follows:
Diagonal Distance = √(Length² + Width²)
= √(16² + 16²)
= √(256 + 256)
= √512
≈ 22.63 meters
Since the diagonal distance is approximately 22.63 meters, which is greater than 14 meters, it means that the layout of the rosebushes, fountain, and garden bench does not meet the homeowner's requirement. The artist needs to adjust the placement to ensure that the distances are no more than 14 meters apart.
B) To determine the coordinates for placing the sundial midway between the fountain and the rosebushes, we can use the concept of the midpoint formula.
The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
From the given information, let's assume the fountain is located at coordinates (x₁, y₁) and the rosebushes are located at coordinates (x₂, y₂).
To find the coordinates of the sundial, we substitute the values into the midpoint formula:
Sundial Coordinates = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Without the specific coordinates provided in the question, we cannot determine the exact coordinates for the sundial. However, by knowing the coordinates of the fountain and the rosebushes, we can use the midpoint formula to calculate the coordinates.