Mathematics
Grade 10
15 min
Estimate products: word problems
Estimate products: word problems
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1
Introduction & Learning Objectives
Learning Objectives
Identify the radius of a circle from its standard equation in the coordinate plane.
Calculate the radius of a circle using the distance formula given the center and a point on the circle.
Apply the formula for the area of a circle (A = πr²) to solve word problems.
Use rounding and mental math to estimate the product of π and other numbers (including r²).
Solve multi-step word problems that require finding a circle's dimensions in a coordinate plane and then estimating a product to find an approximate area or cost.
Justify the reasonableness of an estimated answer in the context of a real-world scenario.
A new cell tower's signal can be modeled by a circle on a map grid. How can you quickly estimate its coverage area in square miles without a cal...
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Key Concepts & Vocabulary
TermDefinitionExample
Standard Equation of a CircleThe formula (x - h)² + (y - k)² = r² which defines a circle with center (h, k) and radius r on the coordinate plane.The equation (x - 2)² + (y + 5)² = 49 represents a circle with its center at (2, -5) and a radius of √49 = 7.
RadiusThe distance from the center of a circle to any point on its circumference.If a circle's center is at the origin (0,0) and it passes through the point (0, 4), its radius is 4 units.
EstimationThe process of finding an approximate value for a calculation, often by rounding numbers to make them easier to work with mentally.To estimate the product of 19.8 and 3.14, you could round them to 20 and 3, respectively, for an estimated product of 60.
ProductThe result of multiplying two or more numbers together.In t...
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Core Formulas
Standard Equation of a Circle
(x - h)^2 + (y - k)^2 = r^2
Use this formula to find the center (h, k) and the radius (r) of a circle. Remember that the value on the right side of the equation is r², so you must take its square root to find the radius.
Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Use this formula to find the distance between two points (x₁, y₁) and (x₂, y₂). In the context of circles, this can be used to find the radius if you know the center and one point on the circle.
Area of a Circle
A = \pi r^2
Use this formula to calculate the area of a circle. The word problems in this lesson will require you to estimate the product of π and the squared radius.
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Challenging
A circular company logo is defined by the equation x² + y² = 50. A square is inscribed inside the circle with its vertices touching the edge. What is the estimated area of the logo that is NOT covered by the square?
A.About 107 square units
B.About 57 square units
C.About 157 square units
D.About 100 square units
Challenging
A circular garden is centered at (0,0) with a radius of 8 meters. A 2-meter wide gravel path is built around the outside of the garden. If gravel costs $25 per square meter, what is the estimated total cost of the gravel for the path?
A.About $5,024
B.About $2,010
C.About $2,826
D.About $816
Challenging
The estimated cost to paint a large circular mural on a wall is $900. The special paint costs about $6 per square foot. Which of the following equations could represent the mural on a coordinate plane?
A.x² + y² = 900
B.x² + y² = 150
C.x² + y² = 25
D.x² + y² = 48
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