Quantitative Reasoning Exam 2
How is angular size related to physical size?
Angular size describes the angle that an object covers as seen from your eye, whereas physical size is an object's true size
unknown sides in similar triangles - T1: a - 15 b - 20 c - 25 - T2: c- 5
x: 3 y: 4
3^x=59
x= 3.71
log10(4+x)=5.2
x=158486.32
The statement does not make sense because both populations grow exponentially, not linearly. 7^x = 53
x=2.04
2^x=55
x=5.78
log 10^x=3.7
x=5011.87
Consider a line with equation y=12x−3. Which of the following lines has the same slope but a different y-intercept? Explain your reasoning.
y=12x+3 because the rate of change is thesame, but the initial value is different.
30 degrees
subtends 1/12 of a circle
90 degrees
subtends to 1/4 of a circle
An air duct in a stadium has a circular cross section with a radius of 15 inches and a length of 42 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?
volume: 206.17 ft^3 duct requires: 329.87 ft^2
An air duct in a stadium has a circular cross section with a radius of 18inches and a length of 46 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?
volume: 325.15 ft^3 painting required: 433.54 ft^2
How is the rate of change of a linear function related to the slope of its graph?
- The smaller the rate of change, the shallower the graph. - The greater the rate of change, the steeper the graph.
Find the angular size of a circular object with a 1-inch diameter viewed from a distance of 4 yards.
0.40 degrees
Find the angular size of a circular object with a 1-inch diameter viewed from a distance of 3yards.
0.53 degrees
On the map (right), the length of each east-west block is 1/10 mile and the length of each north-south block is 1/8 mile. Victoria has to walk from the bus stop to the grocery store. Find the shortest walking distance. Then find the straight-line distance ('as the crow flies') between the two locations.
0.75
Determine which of the following pair of surfaces is steeper. A roof with a pitch of 1 in 4 or a roof with a slope of 2/10
1 in 4 is steeper
A circle with a diameter of 59 feet
185.4 ft - 2734 sq ft
Sally is 6feet tall. She is trying to estimate the height of a nearby building. She makes the following observations. If she stands 15feet away from a lamp post, she can line up the top of the building with the top of the lamp post. Furthermore, she can determine that the top of the lamp post is 4feet above her head and that the building is 50feet from her sighting position. What is the height of the building? Draw a good picture!
19.3 feet
The figure to the right shows the layout of a backyard that is to be seeded with grass except for the patio and flower garden. What is the area of the region that is to be seeded with grass - a: 28m b: 12m - patio: 6m garden 4x4m
215.5 m^2
Suppose someone builds an architectural model of a new office complex using a scale factor of 60 and wants to fill both the scale model office complex and the actual office complex with marbles. How many times the number of marbles required for the model would be required for the actual building?
216000 times as many
If prices increase at a monthly rate of 1.7% what percentage do they increase in a year?
22.4%
The water reservoir for a city is shaped like a rectangular prism 230meters long, 80meters wide, and 13meters deep. At the end of the day, the reservoir is 90%full. How much water must be added overnight to fill the reservoir?
23,920 cubic meters
Suppose you magically quintupled in size. That is, your height, width, and depth increased by a factor of 5. By what factor has the amount of material required for your clothes increased?
25
Log10(5+x)=4.4
25113.86
7/10 circle
252 degrees
A toxic radioactive substance with a density of 8 milligrams per square centimeter is detected in the ventilating ducts of a nuclear processing building that was used 35 years ago. If the half-life of the substance is 20 years, what was the density of the substance when it was deposited 35 years ago?
26.9%
The figure to the right shows the layout of a backyard that is to be seeded with grass except for the patio and flower garden. What is the area of the region that is to be seeded with grass - a: 28m b: 14m - patio: 7m garden 4x4m
299.1m^2 is the seeded region
270 degrees
3/4 circle
An air duct in a stadium has a circular cross section with a radius of 6 inches and a length of 49 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?
38.48ft^3 - 153.94 ft^2
43 S, 122 E
42 N, 58 W
43 N, 108 W
43 S, 72 E
The figure shows a region to be covered with plywood under a set of stairs. The area of that region is 52 square feet. Suppose that the stairs rise at a shallower angle and are 11 feet tall. What is the area of the region to be covered in this case? - 13 ft 8 ft
44 ft ^2
What is the angle (relative to the horizontal) of a 15in 15 roof? Is it possible to have a 18 in 15 roof? Explain.
45 degrees - It is possible to have a 18 in 15 roof. The roof is steep, and some roofs need to be more steep than others
46 S , 104 W
46 N, 76E
Find the latitude and longitude of the location on Earth precisely opposite of a town located at latitude 48°N, longitude 101°W.
48 S 79 E
A slow pitch softball diamond is actually a square 60ft on a side. How far is it from home to second base?
84.9
A city regulates the heights of its houses by the rule that on the shortest day of the year, a house cannot cast a noontime shadow that reaches farther than the shadow that would be cast by a 12-foot fence on the property line. Assume a 12-foot fence casts a 10-foot shadow on the shortest day of the year, and the house is set back 60 feet from the property line.
84ft
A slow pitch softball diamond is actually a square 62 ft on a side. How far is it from home to second base?
87.7 ft
Between 1862and 2008, carbon dioxide (CO2)concentration in the atmosphere rose from roughly 276parts per million to 374parts per million. Assume that this growth can be modeled with an exponential function Q=Q0×(1+r)t. Complete parts (a) and (b) below.
A. By experimenting with various values of the growth rate r, find an exponential function that fits the data for 1862and 2008. - r = 0.00209 B. Use this exponential model to predict when the CO2concentration will double its 1862level. - According to this model, the CO2concentration will double its 1862 lvl in 21952195.
A certain element has a half life of 2.5 billion years. a. You find a rock containing a mixture of the element and lead. You determine that 55% of the original element remains; the other 45% decayed into lead. How old is the rock? b. Analysis of another rock shows that it contains 15% of its original element; the other 85% decayed into lead. How old is the rock?
A. The rock is approximately 2billion years old. B. The rock is approximately 8billion years old.
Annual oil production for a region was about 520million tons in 1950. Production increased at a rate of 7.7%per year between 1950 and 1972,but the rate of growth then slowed. The oil production reached approximately 4.7 billion tons per year in 2016. Complete parts (a) through (d) below.
A. Using the growth rate of 7.7%for the period 1950 to 1972, approximately what was the oil production in 1972 - The oil production is approximately 2.7billion tons in 1972. B. Starting from the result of part (a), approximately how much oil would have been produced in 2016 if growth in production had continued at a rate of 7.7%between 1972and 2016Compare this result to the actual 2016 figure given above - The oil production is approximately 70.6billion tons in 2016. - The actual figure is far less than the figure calculated using 7.7% growth rate. C. Starting from the result of part (a), approximately how much oil would have been produced in 2016 if growth in production had proceeded at a rate of 3.4% between 1972 and 2016. Compare this result to the actual 2016 figure given above. -The oil production is approximately 11.811.8 billion tons in 2016. - The actual figure is far less than the figure calculated using 3.4%growth rate. D. By trial and error, estimate the annual growth rate in oil production between 1972and 2016 with an exponential function. - The annual growth rate in oil production between 1972and 2016 is approximately 1.31%
You can rent time on computers at the local copy center for a $6 setup charge and an additional $1.75 for every 5 minutes. How much time can be rented for $22?
The independent variable is time (t), in minutes, and the dependent variable is rental cost (r), in dollars. The linear function that models this situation is r=0.35t+6. - 46 minutes can be rented for $22. - A linear model is reasonable for this situation.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. My freeway speed is the rate of change in my distance with respect to time.
The statement makes sense because the slope of a linear function is the change in the dependent variable divided by the change in the independent variable.
The amount of sugar in a fermenting batch of beer decreases with time at a rate of 0.2 grams per day, starting from an initial amount of 5 grams. When is the sugar gone?
The sugar is gone after 25 days - It is unknown whether the model is reasonable because a biologist would need to assess the assumption that the sugar consumption rate is constant.
Suppose you draw two parallel lines in a plane. If a third line is perpendicular to one of the two parallel lines, is it necessarily perpendicular to the other line? Explain
The third line would necessarily be perpendicular to both lines.
The figure shows a region to be covered with plywood under a set of stairs. The area of that region is 55 square feet. Suppose that the stairs rise at a shallower angle and are 9 feet tall. What is the area of the region to be covered in this case? a: 10 ft b: 11ft
area: 45ft^2
A circle with a diameter of 45ft
circumference: 141.4 ft area: 1590.4 ft
On the map (right), the length of each east-west block is 1/8 mile and the length of each north-south block is 1/4 mile. Victoria has to walk from the grocery store to the bus stop. Find the shortest walking distance. Then find the straight-line distance ('as the crow flies') between the two locations.
shortest: 1.38 longest: 1.07
72 degrees
subtends to 1/5 of a circle
3/5 circle
subtends to 216 degrees
216 degrees
subtends to 3/5 of a circle
y=-4x+7
the slope is -4 - the y intercept is 7
The diameter of a tree increases by 0.25inch with each passing year. When the tree was first observed, its diameter was 5 inches. Estimate the time at which the tree started growing
the tree started growing 20 yrs ago - It is unknown whether the model is reasonable because the assumption that the diameter grows linearly in time would need to be evaluated by a biologist.
An air duct in a stadium has a circular cross-section with a radius of 24 inches and a length of 33 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?
volume: 414.69 ft^3 painting: 4164.69 ft^2
An air duct in a stadium has a circular cross section with a radius of 24 inches and a length of 37 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?
volume: 464.96 ft^3 duct: 464.96 ft^2
A parking lot is shaped like a parallelogram and bounded on four sides by streets, as shown in the figure to the right. How much asphalt (in square yards) is needed to pave the parking lot? a: 14 yd b: 120 yd
16800yd^2 of asphalt needed
174.37 degrees
174,22',12"
A roof with a pitch of 3 in 10,a roof with a slope of 3/4
3/4 is steeper
1/12 circle
30 degrees
Consider the graph to the right. a. In words, describe the function shown on the graph. b. Find the slope of the graph and express it as a rate of change. c. Briefly discuss the conditions under which a linear function is a realistic model for the given situation.
- According to the function, the distance from home decreases by 900 miles every 6 hours - It is a good model if speed is constant for 6 hours.
A city's population starts at 100,000 people and grows at 3% per year for 7 years. In the general exponential equation Q=Q0×(1+r)t, what is r?
0.03
Find the angular size of a circular object with a 3-inch diameter viewed from a distance of 4 yards.
1.19 degrees
Find the angular size of a circular object with a 5-inch diameter viewed from a distance of 6yards.
1.33 degrees
A triangular lot is 150 ft on one side and has a property line of length 800 ft. Find the area of the lot in acres. (Figure not drawn to scale)
1.35 acres
60 degrees
1/6 circle
105,40',36"
105.68 degrees
110, 20', 36"
110.34
144,40'54"
114.68 degrees
118,20',18"
118.34
44 meters of fence
121 square meters
129.31 degrees
129, 18',36"
The water reservoir for a city is shaped like a rectangular prism 220meters long, 80 meters wide, and 15 meters deep. At the end of the day, the reservoir is 95%full. How much water must be added overnight to fill the reservoir?
13,200 meters
Radioactive uranium-235 has a half-life of about 700 million years. Suppose you find a rock and chemical analysis tells you that only 116 of the rock's original uranium-235 remains. How old is the rock?
2.8 billion years old
13/20 circle
234 degrees
A city regulates the heights of its houses by the rule that on the shortest day of the year, a house cannot cast a noontime shadow that reaches farther than the shadow that would be cast by a 12-foot fence on the property line. Assume a 12-foot fence casts a 20-foot shadow on the shortest day of the year, and the house is set back 30 feet from the property line.
30ft
13/15 circle
312 degrees
72 meters of fence
324 sq meters 413 sq meters The area of the circular enclosure is greater than that of the square enclosure. This suggests that a circle provides a greater area than a square for a fixed perimeter.
37.5 degrees
35,30',0"
36.7 degrees
36, 42', 0"
37.4 degrees
37, 24',0"
39 N, 108 E
39 S, 72 W
39.6 degrees
39,36',0"
A linear function is characterized by what quality? Explain your reasoning.
A constant slope because by definition, a linear function has a constant rate of change and a straight-line graph
Which town would have the steepest slope on a graph showing its population as a function of time? Explain your reasoning.
A town growing at a constant rate of 100 people per year because the rate of change is equal to the slope of a function and the greater the rate of change, the steeper the slope
Consider the graph to the right. a. In words, describe the function shown on the graph. b. Find the slope of the graph and express it as a rate of change. c. Briefly discuss the conditions under which a linear function is a realistic model for the given situation.
According to the function, the distance from home decreases by 500 miles every 6 hours - -83.3 mph - It is a good model if speed is constant for 6hours
What do we mean by Euclidean geometry?
Euclidean geometry is the familiar geometry of lines, angles, and planes.
Give an example of a practical problem that can be solved using similar triangles.
Finding the length of your shadow given your height and the height of a tree that casts a shadow of known length.
If prices increase at a monthly rate of 2.4%,by what percentage do they increase in a year?
Over the span of a year, prices will increase by 32.9%.
What is plane geometry? What does it mean for lines to be perpendicular or parallel in a plane?
Plane geometry is the geometry of two-dimensional objects - Two lines that meet in a right angle (90°)are said to be perpendicular. Two lines in a plane that are everywhere the same distance apart are said to be parallel
The average price of a home in a town was $173,00 in 2007 but home prices are rising by 4%per year
Q=$173,000173,000×(1+0.04) - 0 173,000 1: 179,000 2: 187, 117 3: 194,601 4: 202,386 5: 210,481
The average price of a home in a town was $172,00 in 2007 but home prices are rising by 5%per year
Q=172,000×(1+0.05) - 0 172,000 1: 180,600 2: 189, 630 3: 199,112 4: 209,067 5: 219,520
Suppose you are designing a cardboard box that must have a volume of 125 cubic feet. The cost of the cardboard is $0.19 per square foot. What is the most economical design for the box (one that minimizes the cost), and how much will the material in each box cost?
Rectangular prism with equal height, length, and width - $28.50
Charlie is 60 inches tall. Sam is 17% taller than Charlie but proportioned in exactly the same way. (That is, Sam looks like a larger version of Charlie.) How tall is Sam?
Sam is 70.2 inches tall. (Type an integer or a decimal.)
Squirrels and humans are both mammals that stay warm through metabolism that takes place in the body volume. Mammals must constantly generate internal heat to replace the heat that they lose through the surface area of their skin. In general terms, how does the surface-area-to-volume ratio of a squirrel compare to that of a human being?
Squirrels have a higher surface-area-to-volume ratio than humans
Describe how you can graph an exponential function with the help of doubling time or half-life. What is the general shape of an exponential growth function? What is the shape of an exponential decay function?
Start at the point 0^Q0, where Q0 is the initial value of the function. For an exponentially growing quantity, the value Q of the function is 2^Q0after one doubling time, 4^Q 0 after two doubling times, 8^Q 0 after three doubling times, and so on. Fit a curve between these points. - A steeply rising curve - A falling curve that approaches, but never reaches, the horizontal axis
Which is farther from the North Pole, Sydney, Australia (latitude 33.86°S), or Antananarivo, Madagascar (latitude 18.93°S)? Explain.
Sydney is farther from the North Pole because both cities lie in the southern hemisphere and Sydney's degrees latitude are larger in magnitude than Antananarivo's.
The following situation involves a rate of change that is constant. Write a statement that describes how one variable changes with respect to the other, give the rate of change numerically (with units), and use the rate of change rule to answer any questions. - A 1-degree change (increase or decrease) on the Celsius temperature scale is equivalent to a 95-degree change on the Fahrenheit temperature scale. How much does the Fahrenheit temperature increase if the Celsius temperature increases 8 degrees? How much does the Fahrenheit temperature decrease if the Celsius temperature decreases 21 degrees?
The Fahrenheit temperature varies with respect to the Celsius temperature with a rate of change of 9/5°F per degree C. - What is the increase in Fahrenheit temperature if the Celsius temperature increases by 8°? 14.4°F - What is the decrease in Fahrenheit temperature if the Celsius temperature decreases by 21°? 37.8F
Suppose you magically doubled in size. That is, your height, width, and depth increased by a factor of 2.By what factor has the amount of material required for your clothes increased?
The amount of material required for your clothes has increased by a factor of 4
What would be different about the Sun if you viewed it from Mars (which is farther than Earth from the Sun) instead of from Earth?
The angular size of the sun would be smaller because the distance is larger
A toxic radioactive substance with a density of 4milligrams per square centimeter is detected in the ventilating ducts of a nuclear processing building that was used 45years ago. If the half-life of the substance is 20years, what was the density of the substance when it was deposited 45years ago?
The density of the radioactive substance when it was deposited 45years ago was approximately19.0mg/cm2.
A $760 washing machine in a laundromat is depreciated for tax purposes at a rate of $95 per year. Find a function for the depreciated value of the washing machine as it varies with time. When does the depreciated value reach $0
The equation of the line in slope-intercept form is V=760−95t. - It takes 8 years for the machine to depreciate to $0.
A Campus Republicans fundraiser offers raffle tickets for $14 each. The prize for the raffle is a $350 television set, which must be purchased with the proceeds from the ticket sales. Find a function that gives the profit/loss for the raffle as it varies with the number of tickets sold. How many tickets must be sold for the ticket sales to equal the cost of the prize?
The equation used to model this situation is p=(14×t)−350where p is the profit in dollars and t is the number of tickets sold - 25 tickets must be sold for the ticket sales to equal the cost of the prize.
Suppose someone builds an architectural model of a new office complex using a scale factor of 40. How will the height of the actual office complex compare to the height of the scale model?
The height of the actual building will be 40 times as great as the height of the model.
Suppose someone builds an architectural model of a new concert hall using a scale factor of 65 How will the height of the actual concert hall compare to the height of the scale model?
The height of the actual building will be 65times as great as the height of the model.
The price of a particular model car is $15,000 today and rises with time at a constant rate of $810 per year. How much will a new car of this model cost in 3.3years?
The independent variable is time (t), in years, and the dependent variable is the price (p), in dollars. The linear function that models this situation is p= 15,000+(810×t). - The price of a car after 3.3 years will be $17,67317,673. - The linear model is most likely not reasonable, because it is unlikely that the price of a new car of the same model will increase at a constant rate.
y=4x+6
The slope is 4 - The y-intercept is 6
y=-6x+3
The slope of the line is −6 - .The y-intercept is 3
The maximum speed of a semitrailer truck up a hill varies with the weight of its cargo. With no cargo, it can maintain a maximum speed of 45 miles per hour. With 20 tons of cargo, its maximum speed drops to 40 miles per hour. At what load does a linear model predict a maximum speed of 0 miles per hour?
The speed of the car is 0 when there is 180 of cargo - The model is most likely not reasonable because maximum truck speeds most likely do not decrease linearly with load, but the model may give a rough approximation.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. When I graphed the linear function, it turned out to be a wavy curve.
The statement does not make sense because a linear function has a straight-line graph.
After 100 years, a population growing at a rate of 2% per year will have grown by twice as many people as a population growing at a rate of 1% per year.
The statement does not make sense because both populations grow exponentially, not linearly.
The two highways look like straight lines and they intersect twice.
The statement does not make sense because two straight lines cannot intersect more than once.
In December, it is winter at 70°W and 44°S.
The statement does not make sense because, in December, the sun shines more directly south of the equator. Therefore, it cannot be winter at 44°S.
The water depth in a lake decreases at a rate of 8 inches per week because of evaporation. How much does the water depth change in 9 weeks? In 16weeks?
The water depth varies with respect to time with a rate of change of −8in./week - decreased by 72 decreased by 128
The following situation involves a rate of change that is constant. Write a statement that describes how one variable changes with respect to the other, give the rate of change numerically (with units), and use the rate of change rule to answer any questions. - The water depth in a lake decreases at a rate of 9 inches per week because of evaporation. How much does the water depth change in 8 weeks? In 16 weeks?
The water depth varies with respect to time with a rate of change of −9in./week - How much has the water depth changed after 8 weeks? The water depth has decreased by 72 inch(es). - How much has the water depth changed after 16 weeks The water depth has decreased by 144 inch(es).
Examine a closed book. a. How many dimensions are needed to describe the book? Explain. b. How many dimensions describe the surface (cover) of the book? Explain. c. How many dimensions describe an edge of the book? Explain d. Describe some aspect of the book that represents zero dimensions
Two dimensions are needed because the surface (cover) of the book is a plane object, and two coordinates are needed to locate a point in a plane object. - One dimension is needed because an edge of a book is a line segment, which is a one-dimensional object requiring only one coordinate to locate a point on a line. - A corner of a page A corner of the surface (cover)
unknown sides in similar triangles - T1: a - ? b - 12 c - 27 - T2: b- 8 a - 12 c - ?
a =18 c= 18
y=5x+3
slope: 5 intercept: 3
area of a triangle a = 28 b= 96 c= 100
perimeter: 224 units area: 1344 square units
A sporting goods store sells square exercise mats that are 14m on a side. Find the perimeter and area of the mat.
perimeter: 56m area: 196 m^2
You can rent time on computers at the local copy center for a$7 setup charge and an additional $2.75 for every 5 minutes. How much time can be rented for $23?
r=0.55t+7 - 29 min for $23 - IS reasonable
The figure to the right shows the layout of a backyard that is to be seeded with grass except for the patio and flower garden. What is the area of the region that is to be seeded with grass - a: 28m b: 14m - patio: 7m garden 2x2m
seeded region: 311.1 m^2