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mathematics
calculus

Questions and Answers of Calculus

You are starting a small business. You have 9 investors who are willing to share equally in the venture. When you add 3 more investors, each person's share decreases by $2500. What is the total
You commute 56 miles one way to work. The trip to work takes 10 minutes longer than the trip home. Your average speed on the trip home is 8 miles per hour faster. What is your average speed on the
A car radiator contains 10 liters of a 30% antifreeze solution. How many liters should you replace with pure antifreeze to get a 50% antifreeze solution?
You invest $6000 at 41/2 % and 51/2 % simple interest. During the first year, the two accounts earn $305. Find the initial investment for each account?
1. The volume V of a cone can be calculated by the formula V = 1/3r2h, where r is the radius and h is the height. Solve for h? 2. The kinetic energy E of an object can be calculated by the formula E
Solve the equation using any convenient method? 1. 15 + x − 2x2 = 0 2. 2x2 − x − 28 = 0 3. 6 = 3x2 4. 16x2 = 25
Use the algebraic tests to check for symmetry with respect to both axes and the origin. Then sketch the graph of the equation. 1. y = −3x + 7 2. x = −8 3. x = y2 - 5 4. y = 3x3
A simply supported 20-foot beam supports a uniformly distributed load of 1000 pounds per foot. The bending moment M (in foot-pounds) x feet from one end of the beam is given by M = 500x(20 −
You throw a softball straight up into the air at a velocity of 30 feet per second. You release the softball at a height of 5.8 feet and catch it when it falls back to a height of 6.2 feet. (a) Use
Write the complex number in standard form? 1. 4 + (−9 2. 3 + (−16 3. i2 + 3i 4. −5i + i2
Perform the operation and write the result in standard form. 1. (6 − 4i) + (−9 + i) 2. (7 − 2i) − (3 − 8i) 3. −3i(−2 + 5i) 4. (4 + i)(3 − 10i)
Write the quotient in standard form? 1. 4 / 1 − 2i 2. 6 − 5i / i 3. 3 + 2i / 5 + i 4. 7i / (3 + 2i)2
Perform the operation and write the result in standard form? 1. 4 / 2 - 3i + 2 / 1 + i 2. 1 / 2 + i - 5 / 1 + 4i
Use the Quadratic Formula to solve the quadratic equation? 1. x2 − 2x + 10 = 0 2. x2 + 6x + 34 = 0 3. 4x2 + 4x + 7 = 0 4. 6x2 + 3x + 27 = 0
Solve the equation. Check your solutions? 1. 5x4 − 12x3 = 0 2. 4x3 − 6x2 = 0 3. x3 − 7x2 + 4x − 28 = 0 4. 9x4 + 27x3 − 4x2 − 12x = 0
The demand equation for a hair dryer is p = 42 − (0.001x + 2, where x is the number of units demanded per day and p is the price per unit. Find the demand when the price is set at $29.95?
The average number N (in thousands) of daily newspapers in circulation in the United States from 2000 through 2014 can be approximated by the model N = 56,146 − 314.980t3/2, 0 ( t ( 14, where t
Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded? 1. (−7, 2] 2. (4, () 3. (−(, −10] 4. [−2, 2]
1. An ordered pair (a, b) is a ________ of an equation in x and y when the substitutions x = a and y = b result in a true statement. 2. The set of all solution points of an equation is the ________
Complete the table. Use the resulting solution points to sketch the graph of the equation.1. y = ˆ’2x + 52. y + 1 = 3/4x
Identify the x- and y-intercepts of the graph.1. y = (x - 3)22. y = 16 - 4x2 3. y = |x + 2| 4. y2 = 4 - x 5. y = 2 - 2x3 6. y = x3 - 4x
Use the algebraic tests to check for symmetry with respect to both axes and the origin. 1. x2 − y = 0 2. x − y2 = 0 3. y = x3
Assume that the graph has the given type of symmetry. Complete the graph of the equation. To print an enlarged copy of the graph, go to MathGraphs.com.1.y-axis symmetry 2. x-Axis symmetry 3. Origin
Test for symmetry and graph the equation. Then identify any intercepts. 1. y = −3x + 1 2. y = 2x - 3 3. y = x2 − 2x
Use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. 1. y = 5 - ½ x 2. y = 2/3 x -1 3. y = x2 - 4x +3 4. y = x2 + x -2
Write the standard form of the equation of the circle with the given characteristics. 1. Center: (0, 0); Radius: 3 2. Center: (0, 0); Radius: 7 3. Center: (−4, 5); Radius: 2
Find the center and radius of the circle with the given equation. Then sketch the circle. 1. x2 + y2 = 25 2. x2 + y2 = 36 3. (x − 1)2 + (y + 3)2 = 9 4. x2 + (y − 1)2 = 1
Determine whether each point lies on the graph of the equation. 1. y = √x + 4 (a) (0, 2) (b) (5, 3) 2. y = √5 - x (a) (1, 2) (b) (5, 0)
A hospital purchases a new magnetic resonance imaging (MRI) machine for $1.2 million. The depreciated value y (reduced value) after t years is given by y = 1,200,000 − 80,000t, 0 ≤ t ≤ 10.
You purchase an all-terrain vehicle (ATV) for $9500. The depreciated value y (reduced value) after t years is given by y = 9500 − 1000t, 0 ≤ t ≤ 6. Sketch the graph of the equation.
A regulation NFL playing field of length x and width y has a perimeter of 3462 / 3 or 1040 / 3 yards. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified
1. The arch support of a bridge is modeled by y = −0.0012x2 + 300, where x and y are measured in feet and the x-axis represents the ground. (a) Use a graphing utility to graph the equation. (b)
The table shows the life expectancies of a child (at birth) in the United States for selected years from 1940 through 2010. (Source: U.S. National Center for Health Statistics)A model for the life
The resistance y (in ohms) of 1000 feet of solid copper wire at 68 degrees Fahrenheit isy = 10,370 / x2 where x is the diameter of the wire in mils (0.001 inch).(a) Complete the table.(b) Use the
1. The graph of a linear equation cannot be symmetric with respect to the origin. 2. The graph of a linear equation can have either no x-intercepts or only one x-intercept. 3. A circle can have a
The graph shows the circle with the equation x2 + y2 = 1. Describe the types of symmetry that you observe.
1. Find a and b when the graph of y = ax2 + bx3 is symmetric with respect to (a) the y-axis and (b) the origin. (There are many correct answers.)
1. An ________ is a statement that equates two algebraic expressions. 2. There are three types of equations: ________, ________ equations, and ________. 3. A linear equation in one variable x is an
Solve the equation and check your solution. (If not possible, explain why.) 1. x + 11 = 15 2. 7 − x = 19 3. 7 − 2x = 25 4. 7x + 2 = 23
Solve the equation and check your solution. (If not possible, explain why.) 1. 3x / 8 - 4x / 3 = 4 2. 2x / 5 + 5x = 5 /3 3. 5x / 4 + 1 / 2 = x - 1 / 2 4. x / 5 - x / 2 = 3 + 3x / 10
Find the x-and y-intercepts of the graph of the equation algebraically.1. y = 12 − 5x2. y = 16 − 3x3. y = −3(2x + 1)
Use a graphing utility to graph the equation and approximate any x-intercepts. Then set y = 0 and solve the resulting equation. Compare the result with the graph's x-intercept. 1. y = 2(x − 1) -
Solve the equation. (Round your solution to three decimal places.) 1. 0.275x + 0.725(500 − x) = 300 2. 2.763 − 4.5(2.1x − 5.1432) = 6.32x + 5 3. 2 / 7.398 - 4.405 / x = 1 / x 4. 3 / 6.350 - 6 /
1. The surface area S of the circular cylinder shown in the figure is S = 2Ï€ (25) + 2Ï€ (5h). Find the height h of the cylinder when the surface area is 471 square feet. Use 3.14
The surface area S of the rectangular solid shown in the figure is S = 2(24) + 2(4x) + 2(6x). Find the length x of the solid when the surface area is 248-square centimeters.
A crime scene investigator discovers a femur belonging to an adult human female. The bone is 18inches long. Estimate the height of the female.Use the following information. The relationship between
Officials search a forest for a missing man who is 6feet 3inches tall. They find an adult male femur that is 23inches long. Is it possible that the femur belongs to the missing man?Use the following
The population y (in thousands) of Raleigh, North Carolina, from 2000 to 2014 can be approximated by the model y = 11.09t + 293.4, 0 ‰¤ t ‰¤ 14, where t represents the year,
Determine whether the equation is an identity, a conditional equation, or a contradiction. 1. 3(x − 1) = 3x - 3 2. 2(x + 1) = 2x - 1 3. 2(x − 1) = 3x + 1 4. 4(x + 2) = 2x + 2
The population y (in thousands) of Buffalo, New York, from 2000 to 2014 can be approximated by the model y = ˆ’2.60t + 291.7, 0 ‰¤ t ‰¤ 14, where t represents the
A delivery company has a fleet of vans. The annual operating cost C (in dollars) per van is C = 0.37m + 2600, where m is the number of miles traveled by a van in a year. What number of miles yields
Determine whether the statement is true or false. Justify your answer. 1. The equation x (3 − x) = 10 is a linear equation. 2. The equation 2x + 3 = x is a linear equation. 3. The equation 2(x −
(a) Complete the table.(b) Use the table in part (a) to determine the interval in which the solution of the equation 3.2x ˆ’ 5.8 = 0 is located. Explain. (c) Complete the table. (d) Use the
Are 3x + 2 / 5 = 7 and x + 9 = 20 equivalent equations? Explain.
(a) Use a graphing utility to graph the equation y = 3x − 6.(b) Use the result of part (a) to estimate the x-intercept.(c) Explain how the x-intercept is related to the solution of 3x − 6 = 0, as
Use the information below about a possible tax credit for a family consisting of two adults and two children (see figure). Earned income:ESubsidy (a grant of money):S = 10,000 - ½ E, 0
Consider the linear equation y = ax + b where a and b are real numbers.(a) What is the x-intercept of the graph of the equation when a ≠ 0?(b) What is the y-intercept of the graph of the
Consider the linear equation ax + by = c(a) What is the x-intercept of the graph of the equation when a ≠ 0?(b) What is the y-intercept of the graph of the equation when b ≠ 0?(c) Use your
The sum of two consecutive natural numbers Write an algebraic expression for the verbal description.
The product of two consecutive natural numbers Write an algebraic expression for the verbal description.
The product of two consecutive odd integers, the first of which is 2n - 1 Write an algebraic expression for the verbal description.
The sum of the squares of two consecutive even integers, the first of which is 2n Write an algebraic expression for the verbal description.
The distance a car travels in t hours at a rate of 55 miles per hourWrite an algebraic expression for the verbal description.
The travel time for a plane traveling at a rate of r kilometers per hour for 900 kilometersWrite an algebraic expression for the verbal description.
The amount of acid in x liters of a 20% acid solution Write an algebraic expression for the verbal description.
The sale price of an item with a 33% discount on its list price L Write an algebraic expression for the verbal description.
The area of a triangle with a base that is 16 inches and a height that is h inchesWrite an algebraic expression for the verbal description.
The total cost of producing x units for which the fixed costs are $2500 and the cost per unit is $40 Write an algebraic expression for the verbal description.
The total revenue obtained by selling x units at $12.99 per unit Write an algebraic expression for the verbal description.
1. The discount d is 30% of the list price L. 2. The amount A of water in q quarts of a liquid is 72% of the liquid. Translate the statement into an algebraic expression or equation.
The number N represents p percent of 672. Translate the statement into an algebraic expression or equation.
The sales for this month S2 are 20% greater than the sales from last month S1. Translate the statement into an algebraic expression or equation.
Write an expression for the area of the figure.1.2.
Write a verbal description of the algebraic expression without using the variable. 1. y + 2 2. x - 8 3. t/6 4. 1/3u 5. z-2/3 6. x + 9/5
The sum of two consecutive natural numbers is 525. Find the numbers. Write a mathematical model for the problem and solve.
The sum of three consecutive natural numbers is 804. Find the numbers. Write a mathematical model for the problem and solve.
One positive number is 5 times another number. The difference between the two numbers is 148. Find the numbers. Write a mathematical model for the problem and solve.
One positive number is 1/5 of another number. The difference between the two numbers is 76. Find the numbers. Write a mathematical model for the problem and solve.
Find two consecutive integers whose product is 5 less than the square of the smaller number. Write a mathematical model for the problem and solve.
Find two consecutive natural numbers such that the difference of their reciprocals is 1/4 the reciprocal of the smaller number. Write a mathematical model for the problem and solve.
A salesperson's weekly paycheck is 15% less than a second salesperson's paycheck. The two paychecks total $1125. Find the amount of each paycheck.
The sale price of a swimming pool after a 16.5% discount is $1210.75. Find the original list price of the pool.
A family has annual loan payments equal to 32% of their annual income. During the year, the loan payments total $15,680. What is the family's annual income?
A family has a monthly mortgage payment of $760, which is 16% of their monthly income. What is the family's monthly income?
A rectangular room is 1.5 times as long as it is wide, and its perimeter is 25 meters.(a) Draw a diagram that gives a visual representation of the problem. Let l represent the length and let w
A rectangular picture frame has a perimeter of 3 meters. The height of the frame is 2/3 times its width. Find the dimensions of the picture frame.
To get an A in a course, you must have an average of at least 90% on four tests worth 100 points each. Your scores so far are 87, 92, and 84. What must you score on the fourth test to get an A in the
You are taking a course that has four tests. The first three tests are worth 100 points each and the fourth test is worth 200 points. To get an A in the course, you must have an average of at least
You are driving on a Canadian freeway to a town that is 500 kilometers from your home. After 30 minutes, you pass a freeway exit that you know is 50 kilometers from your home. Assuming that you
A truck driver travels at an average speed of 55 miles per hour on a 200-mile trip to pick up a load of freight. On the return trip (with the truck fully loaded), the average speed is 40 miles per
Light travels at the speed of approximately 3.0 × 108 meters per second. Find the time in minutes required for light to travel from the sun to Earth (an approximate distance of 1.5 × 1011 meters).
Radio waves travel at the same speed as light, approximately 3.0 × 108 meters per second. Find the time required for a radio wave to travel from Mission Control in Houston to NASA astronauts on the
You measure the shadow cast by One Liberty Place in Philadelphia, Pennsylvania, and find that it is 105 feet long. Then you measure the shadow cast by a nearby three-foot post and find that it is 4
You measure a tree's shadow and find that it is 8 meters long. Then you measure the shadow of a nearby two-meter lamppost and find that it is 75centimeters long. (See figure.) How tall is the tree?
A person who is 6 feet tall walks away from a flagpole toward the tip of the shadow of the flagpole. When the person is 30 feet from the flagpole, the tips of the person's shadow and the shadow cast
A person who is 6 feet tall walks away from a 50-foot tower toward the tip of the tower's shadow. At a distance of 32 feet from the tower, the person's shadow begins to emerge beyond the tower's
You invested a total of $12,000 at 4 1/2% and 5% simple interest. During one year, the two accounts earned $580. How much did you invest in each account?
You invested a total of $25,000 at 3% and 4 1/2% simple interest. During one year, the two accounts earned $900. How much did you invest in each account?
A nursery has $40,000 of inventory in dogwood trees and red maple trees. The profit on a dogwood tree is 25% and the profit on a red maple tree is 17%. The profit for the entire stock is 20%. How
An automobile dealer has $600,000 of inventory in minivans and alternative-fueled vehicles. The profit on a minivan is 24% and the profit on an alternative-fueled vehicle is 28%. The profit for the

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