Problem 61 - Algebra A Combined Function

Question

Solve. Cindy Brown, an architect, is drawing plans on grid paper for a circular pool with a fountain in the middle. The paper is marked off in centimeters, and each centimeter represents 1 foot. On the paper, the diameter of the "pool" is 20 centimeters, and "fountain" is the point (0,0) . a. Sketch the architect's drawing. Be sure to label the axes. b. Write an equation that describes the circular pool. c. Cindy plans to place a circle of lights around the fountain such that each light is 5 feet from the fountain. Write an equation for the circle of lights and sketch the circle on your drawing.

Step-by-Step Solution

Answer

a. Draw a circle with center (0,0) and radius 10 cm. b. Equation: \( x^2 + y^2 = 100 \). c. Lights' equation: \( x^2 + y^2 = 25 \). Sketch lights' circle.

Step 1: Sketch the Architect's Drawing

On a grid with labeled axes, plot the center of the pool at the origin (0,0). Draw a circle with a diameter of 20 cm (since each cm represents 1 foot, the diameter is 20 feet), so the radius is half of that, 10 cm. Label the x-axis and y-axis with centimeters.

Step 2: Equation of the Circular Pool

The general equation for a circle centered at the origin is \( x^2 + y^2 = r^2 \), where \( r \) is the radius. Since the radius of the pool is 10 feet, the equation is \( x^2 + y^2 = 100 \).

Step 3: Sketch the Circle of Lights

The lights form another circle centered at the fountain. Since each light is 5 feet from the fountain, the radius of this circle is 5 feet. Draw this circle with a dotted line to differentiate it from the pool circle, with its center also at the origin.

Step 4: Equation for the Circle of Lights

Again, using the circle equation \( x^2 + y^2 = r^2 \), the radius is 5 feet for the lights, so the equation becomes \( x^2 + y^2 = 25 \).

Key Concepts

Understanding Circumference

Circumference is a key concept in geometry and refers to the distance around the outer edge of a circle. It is similar to the perimeter of a polygonal shape. The formula to calculate the circumference of a circle is given by \( C = 2 \pi r \) or alternatively \( C = \pi d \). Here,

• \( r \) represents the radius of the circle, and
• \( d \) represents the diameter.

Using these formulas, you can find out how much material would be needed to surround a circular area, important for tasks like wrapping decorative lights around a pool. Knowing the circumference also helps in determining the length of a circular path someone might walk.

Circle Equations Explained

Circle equations allow us to describe the set of all points that are equidistant from a center point in a coordinate plane. The most common form of a circle's equation is expressed as:\( (x-h)^2 + (y-k)^2 = r^2 \).Here,

• \((h, k)\) are the coordinates of the center of the circle,
• \(r\) is the radius of the circle.

For Cindy Brown’s circular pool effort, since the center is at the origin \((0,0)\), the equation simplifies to \( x^2 + y^2 = r^2 \).This formula simplifies calculations and provides a clear understanding of the circle's size and position. It's particularly useful for architects to ensure accurate renderings in their drawings.

Radius and Diameter Differences

The terms radius and diameter are often used interchangeably, but they represent different things in geometry. The radius of a circle is the distance from the center of the circle to any point along its edge. Conversely, the diameter is the straight-line distance across the circle, passing through the center, which is exactly twice the length of the radius. This relationship can be expressed as:

• Diameter = 2 × Radius
• Radius = Diameter ÷ 2

In the context of Cindy’s pool, knowing the diameter as 20 feet helps in calculating the radius easily as 10 feet, which is crucial for deriving equations or drawing plans.

Locating Circles on the Coordinate Plane

The coordinate plane provides a systematic way to sketch and describe the position of geometric shapes precisely using x and y axes. For circles, this involves understanding how the location of its center and radius translates into a visual representation.

• The center of a circle on a coordinate plane is often represented as \((h, k)\),
• The radius determines how far out the circle extends from this center point.

Cindy's design places the pool at the origin \((0,0)\), making math easier, as any movement or addition is calculated from this point.The coordinate system also helps to accurately place other features—like lights—relative to the central location.

Architectural Drawing: Merging Art and Math

Architectural drawing leverages geometry to transform mathematical concepts into visual plans for real-world construction. By plotting shapes on grid paper, architects like Cindy Brown can create accurate plans that balance aesthetic appeal and structural necessity.

• Such drawings regularly use circles to represent features like pools, round windows, or decorative elements.
• Creating equations based on these circles ensures measurements stay consistent when scaling from paper to full-size constructions.

A successful architectural drawing connects seamlessly with an architect's vision and the mathematical rigor necessary for coherent construction. Ultimately, it is an essential skill that turns numerical data into creative designs.