Tiffany J.

10 years teaching; specialize in math and special education

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Geometry

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Question:

The rope for the school's flagpole broke and has been carried off in the wind. To make an informed decision about what length of rope to tell the maintenance staff to bring, the principal must determine the height of the flagpole. Luckily, it is a sunny day. So, you and a couple geometry classmates go outside and measure your friend, Tom, at 178 cm. Tom's shadow measures 28 cm and the flagpole's shadow measures 240 cm. Calculate the estimated height of the flagpole in feet. Impress the principal with an explanation of how you arrived at this estimation. Your explanation should include a combination of words and algebra and should be backed-up with mathematical theory.

Tiffany J.

Answer:

Since both, the flagpole and Tom are standing at a 90-degree angle to the ground and the sun is shining on them at the same angle, they are forming similar triangles by AA Theorem. The sides of these similar triangles include each object's shadows, each object's heights, and an imaginary hypotenuse stretching from the tip of each object to the tip of its corresponding shadow. By the Triangle Similarity Theorem, we can conclude that Tom's height to shadow-length ratio is proportional to the flagpole's, as shown in the equation below, with $$x$$ representing the height of the flagpole. $$\frac{178}{28}=\frac{x}{240}$$ $$30720=28x$$ $$1097.14=x$$ The height of the flagpole is approximately 1,097.14 cm. Since there is approximately 30.5 cm in one foot, we can divide the height in centimeters by 30.5, which is equivalent to approximately 36 ft.

Pre-Algebra

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Question:

Judy and Tasha both run on their High School's track team. They are on the same team but are competing for a spot on the team's 4x400m relay team. Judy's most recently ran 200m in 35 seconds and Tasha ran 600m in 95 seconds. As the team coach, you must calculate each runner's speed in meters per second to the nearest one-hundredth of a meter and explain which runner is expected to run the 400m dash quicker using both algebra and written words.

Tiffany J.

Answer:

Judy's speed: $$\frac{200m}{35s}=5.71m/s$$ Tasha's speed: $$\frac{600m}{95s}=6.32m/s$$ Judy's expected 400m time: $$\frac{200m}{35s}=\frac{400m}{x}$$ $$\frac{200}{35}=\frac{400}{x}$$ $$200x=14000$$ $$x=70$$ Judy is expected to run 400m in 70 seconds. Tasha's expected 400m time: $$\frac{600m}{95s}=\frac{400m}{x}$$ $$\frac{600}{95}=\frac{400}{x}$$ $$600x=38000$$ $$x=63.33$$ Tasha is expected to run 400m in 63.33 seconds. Tasha is expected to run the 400m relay faster than Judy.

Algebra

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Question:

John and Amber went to the market and each bought peaches and pears. John bought 3 peaches and 9 pears and spent $15.75. Amber bought 8 peaches and 4 pears and spent $12. John claims that he was overcharged because they each bought 12 pieces of fruit, but he was more expensive. Explain the price of each fruit and why each friend paid a fair amount using both algebra and written words.

Tiffany J.

Answer:

Let $$x$$ be the price of each peach. Let $$y$$ be the price of each pear. John's purchase: $$3x+9y=15.75$$ $$8(3x+9y=15.75)$$ $$24x+72y=126$$ Amber's purchase: $$8x+4y=12$$ $$3(8x+4y=12)$$ $$24x+12y=36$$ John's purchase subtract Amber's purchase to solve for $$y$$. $$24x+72y=126$$ $$-(24x+12y=36)$$ $$0x+60y=90$$ $$60y=90$$ $$y=1.5$$ Each pear costs $$$1.50$$. Substitute $$1.5$$ for $$y$$ in an original equation and solve for $$x$$. $$3x+9(1.5)=15.75$$ $$3x+13.5=15.75$$ $$3x=15.75$$ $$x=.75$$ Each peach costs $$$0.75$$. Since John bought more of the more expensive fruit, he ended up paying more overall.

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