Question

Located in Al Raha, Abu Dhabi, the headquarters of property developing company Aldar is a vertically circular building with a diameter of 121 meters. The tip of the building is 110 meters aboveground. Find an equation for the building's outline if the center of the building is on the \(y\) -axis.

Short answer

The equation for the building's outline is \ x^2 + (y - 49.5)^2 = 3660.25.\

Step-by-step solution

Identify the Center and Radius

The building is a circle with a diameter of 121 meters, so the radius is half of that: \[\text{radius} = \frac{121}{2} = 60.5 \text{ meters}\]. Given the tip of the building is 110 meters above ground level, and the center is on the y-axis, assume the center \((0, h)\) is vertically below this point by the radius length. Therefore, \h = 110 - 60.5 = 49.5\.

Write the General Form of a Circle's Equation

The standard form of a circle's equation is: \[(x - a)^2 + (y - b)^2 = r^2\] where \(a, b\) is the center and \(r\) is the radius. In this case, our center \(a, b\) is \(0, 49.5\) and our radius \(r\) is \ 60.5.\.

Substitute Values into the Standard Form

Substitute \(0\) for \(a\), \49.5\ for \(b\), and \60.5\ for \radius\ into the circle formula: \[(x - 0)^2 + (y - 49.5)^2 = (60.5)^2\]. Simplify to get \ x^2 + (y - 49.5)^2 = 3660.25.\

Key concepts

radius calculation

The radius of a circle is an important geometric property. It is the distance from the center of the circle to any point on its circumference. You can calculate the radius if you know the diameter. The diameter is twice the radius and stretches across the circle, passing through the center.

In this exercise with the Aldar HQ, the diameter is given as 121 meters. To find the radius, we divide the diameter by 2. Therefore, the radius is \[\text{Radius} = \frac{121}{2} = 60.5 \text{ meters}\].

This calculation is critical for further finding the equation of the circle. Always start by identifying the radius when dealing with circle equations.

geometry

Geometry involves studying various shapes and their properties. When dealing with circles, understanding terms like radius, diameter, and center is crucial.

In this problem, knowing that the circle is vertically aligned and positioned on the y-axis aids in simplifying the setup. The circle’s center must be located directly beneath the top of the building.

With the building's tip 110 meters above ground and having a radius of 60.5 meters, we calculate the center's vertical placement. Subtracting the radius from the top height gives us: \[\text{Center} = 110 - 60.5 = 49.5 \text{ meters}\].

This places the center of the circle at (0, 49.5) on the y-axis, simplifying our equation.

standard form of a circle

The standard form of a circle's equation is a fundamental concept in geometry. It is shown as: \[(x - a)^2 + (y - b)^2 = r^2\]

Here, \((a, b)\) represents the center coordinates, and \(\text{r}\) represents the radius of the circle.

Given the Aldar HQ problem:

• The center \((a, b) = (0, 49.5)\) meters
• The radius \(\text{r} = 60.5\) meters

We substitute these values into the standard form:

\[(x - 0)^2 + (y - 49.5)^2 = (60.5)^2\]

Simplify this to get the final equation:

\[x^2 + (y - 49.5)^2 = 3660.25\]

This resulting equation represents the outline of the circular building.