Geometry Unit 1 Review Problems

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Questions and Answers

What was the percentage decrease in the population of Bristol over the last thirty years?

  • 6.00% (correct)
  • 8.29%
  • 7.01%
  • 5.64%

What equation can represent the relationship between the original price ($p$) and discounted price ($d$) with a 60% discount?

  • $d = 0.6p$
  • $d = p + 0.6$
  • $d = 0.2p$
  • $d = 0.4p$ (correct)

What will the expected cost of the new Des-car be next year if it costs $17,000 this year and the expected increase is 2.9%?

  • $17,329.30
  • $17,000.58
  • $17,497.30 (correct)
  • $17,300.00

If each can of paint covers 15% of Dalia's room, how many cans will she need to paint her entire room?

<p>8 cans (B)</p> Signup and view all the answers

If Dalia has already painted 20% of her room using 75% of a can of paint, how much more paint does she need to finish?

<p>1.5 cans (B)</p> Signup and view all the answers

What is the closest approximation of the circumference of a circle with a radius of 40 centimeters?

<p>251 centimeters (B)</p> Signup and view all the answers

Which order from least accurate to most accurate is correct for the area of a circle with a radius of 6 units calculated by three students?

<p>113.1 sq.units, 113.1Ï€ sq.units, 36Ï€ sq.units (C)</p> Signup and view all the answers

Which statement about the diameter of the wheel is true if the diameter is approximately 31.5 inches?

<p>The relationship in the table appears to be proportional. (B)</p> Signup and view all the answers

What does the area of a circular window correspond to among the following?

<p>Area (D)</p> Signup and view all the answers

In which situation would circumference be relevant?

<p>The distance a toy car travels around a circular track. (B)</p> Signup and view all the answers

Which measurement is always proportional to the diameter of a circle?

<p>Circumference (A)</p> Signup and view all the answers

How is the area of a circle with a diameter of 10 inches calculated?

<p>Using $A = \pi r^2$ with $r = 5$ units. (C)</p> Signup and view all the answers

Which of the following best describes the relationship between the distance traveled and the number of rotations of a wheel?

<p>The relationship is proportional as distance increases with more rotations. (A)</p> Signup and view all the answers

What is the area of Pool A with a 6-foot radius?

<p>113.04 square feet (D)</p> Signup and view all the answers

How much area does one box of tile cover?

<p>25 square feet (A)</p> Signup and view all the answers

How many boxes of tile are needed to cover the bottom of Pool B, which has a 12-foot radius?

<p>8 (B)</p> Signup and view all the answers

If Charlie's pond has a diameter of 6.3 meters, what is the circumference around the pond?

<p>19.60 meters (B)</p> Signup and view all the answers

What is the width of the sidewalk that Charlie wants to build around his pond?

<p>0.8 meters (C)</p> Signup and view all the answers

If a pencil is 120 millimeters long and a marker is 9% longer, how long is the marker?

<p>130.8 mm (A)</p> Signup and view all the answers

What is an accurate estimate for the total amount of gas needed to mow the entire field, if 1/8 of the field requires 1/2 gallon?

<p>4 gallons (C)</p> Signup and view all the answers

What is the total cost Na'ilah will pay for the rug after applying the sales tax and discount?

<p>$53 (D)</p> Signup and view all the answers

If segment GH is three times longer than segment AB, what can be concluded about the relationship between polygons EFGH and ABCD?

<p>EFGH is a scaled copy of ABCD. (B)</p> Signup and view all the answers

What is the area of the rectangular park if it is 330 yards long and the drawing is 11 inches long?

<p>2,640 square yards (C)</p> Signup and view all the answers

Which statement is true when comparing the old map (scale 1 cm to 400 m) and the new map (scale 1 cm to 100 m)?

<p>The new map is larger than the old map. (A)</p> Signup and view all the answers

If the scale factor from polygon EFGH to ABCD is 3, what is the relationship of their areas?

<p>The area of EFGH is 9 times the area of ABCD. (B)</p> Signup and view all the answers

What is the actual distance between the school and the playground if they are 8 centimeters apart on the map, assuming a scale of 1 cm to 5 kilometers?

<p>20 kilometers (A)</p> Signup and view all the answers

In which scenario would the mapped distance between two locations be represented accurately?

<p>The map is proportional and uses a consistent scale throughout. (D)</p> Signup and view all the answers

If a trail is 20 cm long on the old map, how long is this trail on the new map that has a scale of 1 cm to 100 m?

<p>200 m (C)</p> Signup and view all the answers

What conclusion can be drawn from a graph representing a proportional relationship?

<p>The line passes through the origin. (C)</p> Signup and view all the answers

Which statement correctly identifies a true proportional relationship about strawberries?

<p>10 pounds of strawberries cost $32. (C)</p> Signup and view all the answers

What is the correct equation for the time it takes Abdullah to walk a distance of $d$ miles?

<p>$t = \frac{d}{15}$ (C)</p> Signup and view all the answers

Which runner, based on the distance they ran, can be identified as running more slowly?

<p>Runner E ran at a slower speed. (D)</p> Signup and view all the answers

Which scenario represents a proportional relationship between the number of blueberries and their weight?

<p>400 blueberries weigh 2.00 kg. (D)</p> Signup and view all the answers

What is the meaning of the point (1, 1.25) in the context of the recipe for tortillas?

<p>1 cup of flour corresponds to 1.25 cups of water. (A)</p> Signup and view all the answers

Is there a proportional relationship between the mass in pounds and mass in kilograms in the equation $p = 2.2x$?

<p>Yes, because it has a constant ratio. (B)</p> Signup and view all the answers

What would be the correct proportion for each column in the height and running speed table?

<p>Height increases with speed but is not proportional. (C)</p> Signup and view all the answers

If 5 cups of flour are used for every 4 cups of water, which ratio best describes the relationship?

<p>Flour to water ratio is 5:4. (D)</p> Signup and view all the answers

Flashcards

Circumference

The distance around a circle.

Area

The space inside a circle.

Diameter

The distance across a circle through the center.

Radius

The distance from the center of a circle to the edge.

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Proportional

The relationship between two quantities where their ratio is constant.

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Circumference & Diameter

The circumference of a circle is always proportional to its diameter.

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Area & Radius

The area of a circle is proportional to the square of its radius.

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Circumference Formula

The formula to calculate the circumference of a circle is C=πd, where C is the circumference, π is pi (approximately 3.14), and d is the diameter.

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Perimeter

The total length of the outside boundary of a shape.

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Percent

A way to represent part of a whole as a fraction.

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Percent Error

The error in a measurement compared to the actual value, expressed as a percentage.

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Sales Tax

A sales tax is a percentage added to the original price of an item.

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Discount

A discount is a percentage taken off the original price of an item.

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Percentage Decrease

The percentage decrease in a quantity is calculated by dividing the amount of change by the original quantity and multiplying by 100%.

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Proportional Relationship

A proportional relationship exists when two quantities change at a constant rate. This means that their ratio remains the same.

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Discount Equation

An equation that represents a proportional relationship between two variables, with the constant of proportionality (discount) represented by a coefficient. In this case, the discount price is equal to the original price minus the discount percentage of the original price.

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Percent Increase

The increase in price from one year to the next, expressed as a percentage of the original price.

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Compound Increase

When a quantity grows at a constant percentage rate over a period of time, each increase is calculated based on the previous year's value. This creates a compound effect, where the increase gets larger with each passing year.

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What is a scaled copy?

A scaled copy is an image that has been enlarged or reduced proportionally. The scale factor tells you how much bigger or smaller the scaled copy is compared to the original. For example, a scale factor of 2 would mean the scaled copy is two times bigger than the original.

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What is the relationship between side lengths in scaled copies?

When two figures are scaled copies, their corresponding side lengths are proportional. This means the ratio of any two corresponding sides is the same. For example, if the scale factor is 3, then the sides of the scaled copy will be three times longer than the sides of the original.

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How does the scale factor affect the area of a scaled copy?

The area of a scaled copy changes by the square of the scale factor. If the scale factor is 2, the area of the scaled copy will be 22 = 4 times bigger than the original. If the scale factor is 1/2, the area of the scaled copy will be 1/21/2 = 1/4 times smaller than the original.

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What is a proportional relationship?

A proportional relationship is a relationship between two quantities where the ratio between them is constant. This means that as one quantity increases, the other quantity also increases proportionally.

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How is a proportional relationship represented on a graph?

On a graph, a proportional relationship is represented by a straight line that passes through the origin (0,0). This line shows that as one quantity increases, the other quantity increases at a constant rate.

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How can you tell if a graph represents a proportional relationship?

A graph represents a proportional relationship when the ratio between the two quantities is constant. This means the line representing the relationship is straight and passes through the origin.

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Direct Proportion

A relationship where the output is a constant multiple of the input. You can express this as y = kx, where k is the constant of proportionality.

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Abdullah's Walk Equation

The equation that represents the relationship between time (t) and distance (d) for Abdullah's walk is t = 15d. This means the time taken is directly proportional to the distance covered, with the constant of proportionality being 15, representing the time per unit distance.

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Runner's Speed

Identifying which of the runners ran more slowly requires analyzing the slope of the lines representing their distances. The runner with a shallower slope ran more slowly because their distance increased less rapidly over the same time period.

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Blueberries Weight Proportion

A proportional relationship exists between the number of blueberries and their weight if the ratio of weight to number is constant. To complete the table, we need to find the constant ratio (weight per blueberry) and apply it to the other values.

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Mass Conversion Proportion

The equation p = 2.22x represents a proportional relationship between mass in pounds (p) and mass in kilograms (x) because it follows the form p = kx, where k (2.22) is the constant of proportionality.

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Flour and Water Equation

The equation that represents the proportional relationship between flour (f) and water (w) is f = 1.25w. This equation shows that for every cup of water, you need 1.25 cups of flour.

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Point on the Line Meaning

The point (1, 1.25) represents the situation where 1 cup of water requires 1.25 cups of flour, according to the recipe.

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Study Notes

Unit 1 Review Problems

  • Rectangle Scaling: Identify scaled copies of rectangle A. Multiple rectangles are shown.

Polygon Scaling

  • Scaled Copies: Polygon EFGH is a scaled copy of polygon ABCD. True statements about the scale factor and relationships between corresponding segments are selected.
  • Scale Factor: The scale factor from EFGH to ABCD is given as 3/5.
  • Segment Lengths: The length of segment BC is not explicitly given but implied by corresponding segment relationships.
  • Area Scaling: The area of EFGH is three times the area of ABCD.

Scale Drawings and Area

  • Scale Drawing Dimensions: A rectangular park is represented in a scale drawing. Dimensions are 8 inches wide and 11 inches long on the map. The actual length of the park is 330 yards.
  • Actual Area: Calculate the actual area of the park in square yards. The correct answer is 79 200 square yards.

Scaling Polygons

  • Scale Factor: Draw a scaled copy of a polygon using a scale factor of 1/4.

Distance and Scale

  • Map Distance to Real Distance (Hospital/School): Calculate the map distance between a school and a hospital that are 15 kilometers apart. A scale of 2 cm equals 5 km is used. The distance between them on the map is asked for.
  • Map Distance to Real Distance (School/Playground): Calculate the actual distance between a school and a playground when they are 8 centimeters apart on a map with a scale of 2 cm to 5 km.

Map Scale and Size

  • Map Size Relationship: A trail runner has two maps of a mountain. Her old map scale is 1cm to 400 m, and her new map scale is 1 cm to 100 m. If the maps represent the same area, how will the new map compare in size to the old map?
  • Trail Length (Old to New Maps): A trail on the old map was 20 cm long. What is the length of the trail on the new map?

Campus Map and Distances

  • Campus Map Scale: A campus map has a scale of 5 cm to 200 m. Distances (in cm) are provided for segments on the map between buildings.
  • Distance Calculations: Calculate the distances between various buildings (Clarkson Hall to Barge Commons, Barge Commons to Pressly Hall, and Clarkson Hall to Pressly Hall) in meters, using the scale.

Proportional Relationships

  • Graph Types: Identify the graph that represents a proportional relationship. Multiple graphs are provided.
  • Cost of Strawberries (Graph): A graph shows the cost of strawberries in pounds. True statements related to the price/quantity are listed. A specific point is mentioned.
  • Constant Speed: An equation representing constant speed (1 mile in 15 minutes) that relates distance to time is needed.
  • Runner Comparison: Determine which of two runners ran more slowly given a graph showing distance vs. time.

Proportional Relationship (Weight)

  • Blueberry Weight Table: A table showing the relationship between the number of blueberries and their weight needs to be completed so that the relationship is proportional.

Tortillas (Recipe)

  • Proportional Relationship (Recipe): A recipe for tortillas calls for 5 cups of flour for every 4 cups of water. Graph the relationship. A point is given (1, 1.25). Label the axes. The equation representing this relationship needs to be determined, and the meaning of (1, 1.25) is explained.

Height vs. Running Speed

  • Proportionality: Determine if there is a proportional relationship between height and running speed based on the information in a table. Explain your answer.

Circle Circumference

  • Circle Circumference Calculation: Calculate the circumference of a circle with a given radius. Multiple choices of circumference values are given.

Circle Area

  • Circle Area Calculation: A circle with a radius of 6 units has its area calculations compared (in terms of most & least accurate).
  • Wheel Rotations: Calculate wheel circumference/diameter and state true statements (relationships in a table).

Circle Perimeter/Area

  • Combined Shape: Calculate the perimeter and area of a shape that is made of part of a circle and a rectangle.

Pool Tiling

  • Pool Area (Pool A): Calculate the area (in square feet) of a circular pool with a given radius. This area represents the amount of tile that is needed.
  • Pool B Comparison: A second pool (Pool B) has a larger radius. Determine whether the amount of tile needed for Pool B is four times as much as Pool A's tile.

Concrete Sidewalk

  • Concrete Sidewalk Distance: Calculate the distance around a circular fish pond (including a sidewalk). The radius of the fish pond and the sidewalk width are given.
  • Concrete Area: Calculate the area of the concrete needed for the sidewalk.

Pencil/Marker Length

  • Percent Increase: A pencil is 120 mm long. A marker is 9% longer. Determine the length of the marker.

Gas Consumption

  • Gas for Mowing a Field: Kayleen is mowing a field. 1/8 of the field is mowed, and 1/2 gallon of gas is used. Find the total number of gallons required to mow the entire field.

Temperature Error

  • Temperature Error: The temperature in a room is 20.0°C. A thermometer measures it as 20.8°C. Find the percent error of this measurement.

Rug Purchase

  • Sales Tax and Coupon: Na'ilah buys a rug that costs $50. A 10% sales tax and a subsequent 10% off coupon are applied. Find the final cost.

Population Change

  • Population Decrease: The population of Bristol, PA is currently 53,110. Thirty years ago the population was 56,500. Calculate the percentage decrease between the two values.

Discount Price

  • Discount Relationship: Determine if a 60% discount on items is proportional to the original price of the item. Explain your answer.
  • Discount Price Equation: Write an equation for the relationship between an item's discount price "d" and its original price "p".

Car Value Increase

  • Car Price Increase: The price of a Des-car is expected to increase by 2.9% next year. The current cost is $17,000. Find the expected price next year.
  • Car Price Increase Equation: Write an equation representing the increase in car cost and specify variables used (current year cost and next year's cost)
  • Future Car Value: Find the expected cost of a Des-truck (currently $22,000) in two years if its percent increase stays constant. Explain your reasoning.

Paint Usage

  • Percentage of Paint (Room): Helen is painting her room. She used 75% of a can of paint after painting 20% of her room. Identify the true statements related to the total paint needed/the percentage of the room.

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