A floor plan for a house is drawn with a scale of 41 inch = 1 foot. The rectangular living room on the plan measures 5 inches by 4.5 inches. If carpeting costs $3.00 per square foot, what will be the total cost to carpet the living room?
Answer: $1,080.00
The scale 41 inch = 1 foot means that 1 inch = 4 feet. Convert the plan dimensions to actual dimensions: Length = 5 inches 4 ft/inch = 20 feet. Width = 4.5 inches 4 ft/inch = 18 feet. Calculate the actual area: Area = 20 ft 18 ft = 360 square feet. Finally, calculate the cost: Cost = 360 sq ft $3.00/sq ft = $1,080.00.
A car's gas tank holds 15 gallons. The car gets an average of 30 miles per gallon. The driver starts a 400-mile trip with a full tank of gas. Which statement is the most accurate estimation of the situation at the end of the trip?
Answer: The driver will have about 1.7 gallons of gas left.
First, calculate the car's maximum range on a full tank: 15 gallons * 30 miles/gallon = 450 miles. Since the trip is 400 miles long, the car can make the trip without refueling. Next, calculate how much gas will be used: 400 miles / 30 miles/gallon ≈ 13.33 gallons. To find the remaining gas, subtract the used gas from the full tank: 15 gallons - 13.33 gallons = 1.67 gallons. The best estimate is that the driver will have about 1.7 gallons left.
A city planner is reviewing a map with a scale of 1 inch = 0.5 miles. A new bike path is planned that will run in a straight line between two points that are 4.5 inches apart on the map. If a cyclist travels at an average speed of 9 miles per hour, how many minutes will it take to travel the length of the new path?
Answer: 15 minutes
Step 1: Find the actual distance of the bike path. Distance = 4.5 inches 0.5 miles/inch = 2.25 miles. Step 2: Calculate the travel time in hours. Time = Distance / Speed = 2.25 miles / 9 mph = 0.25 hours. Step 3: Convert the travel time to minutes. Time in minutes = 0.25 hours 60 minutes/hour = 15 minutes.
A blueprint shows a rectangular garden with dimensions 8 inches by 12 inches. The scale of the blueprint is such that 1 square inch of blueprint area represents 9 square feet of actual garden area. What is the actual perimeter of the garden in feet?
Answer: 120 feet
The area scale is 1 sq in : 9 sq ft. To find the linear scale, take the square root of the area scale factor: 1 in : 9 ft, which is 1 inch : 3 feet. Now, use the linear scale to find the actual dimensions. Length = 12 in 3 ft/in = 36 ft. Width = 8 in 3 ft/in = 24 ft. The actual perimeter is 2 (Length + Width) = 2 (36 ft + 24 ft) = 2 * 60 ft = 120 feet.
On a blueprint with a scale of 1:120, the length of a rectangular room is 2.5 inches. The actual width of the room is 20 feet. What is the width of the room on the blueprint?
Answer: 2.0 inches
The scale 1:120 means 1 inch on the blueprint represents 120 inches in reality. The actual width of the room is 20 feet. First, convert this to inches: 20 feet * 12 inches/foot = 240 inches. To find the width on the blueprint, divide the actual width by the scale factor: Blueprint Width = 240 inches / 120 = 2.0 inches. The length of the room is extra information not needed to solve the problem.
A scale drawing of a triangular park has side lengths of 6 cm, 8 cm, and 10 cm. The scale of the drawing is 1 cm = 15 meters. What is the actual perimeter of the park in meters?
Answer: 360 meters
There are two ways to solve this. Method 1: Convert each side length to its actual size first. Side 1 = 6 cm 15 m/cm = 90 m. Side 2 = 8 cm 15 m/cm = 120 m. Side 3 = 10 cm 15 m/cm = 150 m. The actual perimeter is the sum of the actual side lengths: 90 + 120 + 150 = 360 meters. Method 2: Find the perimeter on the drawing first: 6 + 8 + 10 = 24 cm. Then, convert this perimeter to the actual perimeter using the scale: 24 cm 15 m/cm = 360 meters.
A scale model of a shipping container is built with a scale of 1:50. The model has dimensions of 3 inches long, 2.4 inches wide, and 1.8 inches high. What is the volume of the actual shipping container in cubic feet? (1 foot = 12 inches)
Answer: 937.5 cubic feet
First, find the actual dimensions by multiplying the model's dimensions by the scale factor of 50. Actual length = 3 in 50 = 150 in. Actual width = 2.4 in 50 = 120 in. Actual height = 1.8 in 50 = 90 in. Next, convert these dimensions to feet. Length = 150 in / 12 in/ft = 12.5 ft. Width = 120 in / 12 in/ft = 10 ft. Height = 90 in / 12 in/ft = 7.5 ft. Finally, calculate the volume: V = 12.5 ft 10 ft * 7.5 ft = 937.5 cubic feet.
The length of a football field is 100 yards. A television screen showing the field is 20 inches wide. If the full length of the field is shown across the width of the screen, which of the following is the best estimate for the scale of the image on the screen? (1 yard = 3 feet, 1 foot = 12 inches)
Answer: 1 inch = 5 yards
First, find the scale by setting the screen width equal to the field length: 20 inches = 100 yards. To find the scale in the form '1 inch = x yards', divide both sides by 20: (20/20) inches = (100/20) yards. This simplifies to 1 inch = 5 yards. Converting to inches (1 yard = 36 inches, so 5 yards = 180 inches) would give a scale of 1:180, but the answer choices are in 'inch to yard' format.
A contractor estimates that a construction project will require 45 cubic yards of concrete. The concrete mixer truck can hold 8 cubic yards. The contractor wants to be sure to have enough concrete. Which of the following represents a reasonable number of trucks to order?
Answer: 6 trucks
To determine the number of trucks needed, divide the total required concrete by the capacity of one truck: 45 cubic yards / 8 cubic yards/truck = 5.625 trucks. Since it is impossible to order a fraction of a truck, the contractor must round up to the next whole number to ensure there is enough concrete for the job. Therefore, the contractor should order 6 trucks.
To estimate the fish population in a lake, researchers catch, tag, and release 200 fish. A week later, they catch a new sample of 150 fish, of which 10 are tagged. Based on this sample, what is the best estimate for the total number of fish in the lake?
Answer: 3,000
This is a capture-recapture estimation problem. Set up a proportion where the ratio of tagged fish to the total population is assumed to be the same as the ratio of tagged fish in the sample to the sample size. Let P be the total population. (Tagged fish in population) / (Total population) = (Tagged fish in sample) / (Sample size). So, 200 / P = 10 / 150. Cross-multiply to solve for P: 10 P = 200 150. 10P = 30,000. P = 3,000.
To estimate the number of words in a 240-page book, a student counts the words on 5 randomly selected pages. The counts are 354, 341, 360, 338, and 352. Which of the following is the BEST estimate for the total number of words in the book?
Answer: 84,000
To find the best estimate, first calculate the average number of words per page from the sample: (354 + 341 + 360 + 338 + 352) / 5 = 1745 / 5 = 349 words per page. Then, multiply this average by the total number of pages: 349 words/page * 240 pages = 83,760 words. The closest rounded estimate is 84,000.
A baker is making a giant cookie that is geometrically similar to a standard cookie. The giant cookie's diameter is 5 times the standard cookie's diameter. If the standard cookie requires 2 ounces of dough, approximately how much dough is needed for the giant cookie?
Answer: 50 ounces
The amount of dough needed is related to the volume of the cookie. Assuming the thickness scales with the diameter (or is a ratio of it), the volume scales by the cube of the linear scale factor. However, cookies are typically flat, so their volume is primarily a function of their area and a relatively constant thickness. The area scales with the square of the linear scale factor. The linear scale factor for the diameter is 5. Therefore, the area (and thus the amount of dough) will scale by 52 = 25. Amount of dough = 2 ounces * 25 = 50 ounces.
On a map, the scale is 1 inch = 2.5 miles. A rectangular state park measures 3.2 inches by 1.6 inches on the map. What is the actual area of the park in square miles?
Answer: 32.0 square miles
First, convert the map dimensions to actual dimensions using the scale. The length is 3.2 inches 2.5 miles/inch = 8 miles. The width is 1.6 inches 2.5 miles/inch = 4 miles. The actual area is the product of the actual dimensions: Area = 8 miles 4 miles = 32 square miles. Alternatively, find the area on the map (3.2 1.6 = 5.12 sq inches) and multiply by the square of the scale factor (2.52 = 6.25). 5.12 * 6.25 = 32 square miles.
On a trail map with a scale of 1 inch to 1.5 miles, a hiking loop measures 8 inches in total length. A hiker estimates they can maintain an average speed of 2 miles per hour, including breaks. Which of the following is the best estimate for the time it will take to complete the hiking loop?
Answer: 6 hours
First, calculate the actual length of the trail using the map scale: Distance = 8 inches * 1.5 miles/inch = 12 miles. Next, use the formula Time = Distance / Speed to estimate the hiking time: Time = 12 miles / 2 miles per hour = 6 hours.
A landscape designer is creating a scale model of a garden. The actual garden has a rectangular flowerbed that is 20 feet long and 8 feet wide. If the designer uses a scale of 1 inch to 4 feet, what is the area of the flowerbed in the model?
Answer: 10 square inches
First, convert the actual dimensions of the flowerbed to the model's dimensions using the scale. Model Length = 20 feet / (4 feet/inch) = 5 inches. Model Width = 8 feet / (4 feet/inch) = 2 inches. Next, calculate the area of the flowerbed in the model: Model Area = Model Length Model Width = 5 inches 2 inches = 10 square inches.
A circular garden with a radius of 14 feet is drawn on a plan with a scale of 1 inch = 4 feet. What is the area of the garden on the plan? (Use π≈722)
Answer: 38.5 square inches
First, find the radius of the garden on the plan. Use the scale to convert the actual radius to the plan radius: Plan Radius = Actual Radius / Scale Factor = 14 feet / (4 feet/inch) = 3.5 inches. Now, calculate the area on the plan using the formula A = πr2. A = π∗(3.5)2=12.25π square inches. Using the approximation π≈722, the area is 722∗12.25=722∗449=211∗7=277=38.5 square inches.
A photograph of a cell is magnified by a factor of 400. In the photograph, the cell's nucleus appears to have a diameter of 8 millimeters. What is the actual diameter of the nucleus in millimeters?
Answer: 0.02 mm
The magnification factor is the scale. The image size is the actual size multiplied by the magnification factor. To find the actual size, we must divide the image size by the magnification factor. Actual Diameter = Image Diameter / Magnification Factor = 8 mm / 400 = 2 / 100 = 0.02 mm.
A map uses a scale where 21 inch represents 43 mile. A road on the map is represented by a line segment that is 5 inches long. What is the actual length of the road?
Answer: 7.5 miles
First, determine the scale factor in miles per inch. If 0.5 inches = 0.75 miles, then 1 inch = 0.75 / 0.5 = 1.5 miles. Now, multiply the map length by this scale factor to find the actual length: Actual Length = 5 inches * 1.5 miles/inch = 7.5 miles.
A cube made of a certain metal has a side length of 2 cm and a mass of 64 grams. A second, larger cube is made of the same metal and has a side length of 5 cm. What is the mass of the larger cube?
Answer: 1000 grams
Mass is proportional to volume for objects made of the same material. First, find the volume of each cube. Volume of small cube = (2cm)3 = 8 cm3. Volume of large cube = (5cm)3 = 125 cm3. Now find the density of the metal using the small cube: Density = Mass / Volume = 64 g / 8 cm3 = 8 g/cm3. Finally, use the density to find the mass of the large cube: Mass = Density * Volume = 8 g/cm3 * 125 cm3 = 1000 grams.
A geographical map uses a scale of 1:250,000. The distance on the map between two cities is 12 centimeters. What is the actual distance between the two cities in kilometers? (1 km = 100,000 cm)
Answer: 30 km
The scale 1:250,000 means 1 cm on the map represents 250,000 cm in reality. First, find the actual distance in centimeters: 12 cm * 250,000 = 3,000,000 cm. Next, convert this distance to kilometers by dividing by 100,000 cm/km: 3,000,000 cm / 100,000 cm/km = 30 km.
