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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Parking at an airport costs $3.00 for the first hour, or any part thereof, and $2.50 for each additional hour, or any part thereof. What range of hours costs at least $28 and no more than $78?
Answer the given questions about the inequality 0 < a < b.What is wrong with the following sequence of steps? a < b, ab < b2, ab − b2 < 0, b(a − b) < 0, b < 0
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y > 8 + 7x − x2
For what values of real numbers a and b does the inequality (x − a)(x − b) < 0 have real solutions?
Draw a graph of the solution of the system y ≥ 2x2 − 6 and y = x − 3.
A contractor is considering two similar jobs, each of which is estimated to take n hours to complete. One pays $350 plus $15 per hour, and the other pays $25 per hour. For what values of n will the
Solve the given problems. Write the relationship between (|x| + |y|) and |x + y| if x > 0 and y < 0.
Solve the given problems. Write the relationship between |xy| and |x||y| if x > 0 and y < 0.
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y < x3 + 4x2 − x − 4
Algebraically find the intervals for which f (x) = 2x4 − 5x3 + 3x2 is positive and those for which it is negative. Using only this information, draw a rough sketch of the graph of the function.
Draw a graph of the solution of the system y < |x + 2| and y = x2.
In designing plastic pipe, if the inner radius r is increased by 5.00 cm, and the inner cross-sectional area is increased by between 125 cm2 and 175 cm2, what are the possible inner radii of the pipe?
Explain the error in the following “proof” that 3 < 2:(1) 1/8 < 1/4 (2) 0.53 < 0.52 (3) log 0.53 < log 0.52(4) 3 log 0.5 < 2 log 0.5 (5) 3 < 2
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y < 32x − x4
The power p (in W) used by a motor is given by p = 9 + 5t − t2, where t is the time (in min). For what values of t is the power greater than 15 W?
A telephone company is installing two types of fiberoptic cable in an area. It is estimated that no more than 300 m of type A cable, and at least 200 m but no more than 400 m of type B cable, are
The relation between the temperature in degrees Fahrenheit F and degrees Celsius C is 9C = 5(F − 32). What temperatures F correspond to temperatures between 10°C and 20°C?
The weekly sales S (in thousands of units) t weeks after a new smart-watch is released on the market is given byWhen will the sales be 4000 units or more? S = 100t t² + 100 2
Solve the given problems. If x ≠ y, show that x2 + y2 > 2xy.
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y > 2x – 1y < 6 – 3x2
The weight w (in tons) of fuel in a rocket after launch is w = 2000 − t2 − 140t, where t is the time (in min). During what period of time is the weight of fuel greater than 500 tons?
A refinery can produce gasoline and diesel fuel, in amounts of any combination, except that equipment restricts total production to 2.5 × 105 barrels day. Graph the different possible production
The voltage drop V across a resistor is the product of the current i (in A) and the resistance R (in Ω). Find the possible voltage drops across a variable resistor R, if the minimum and maximum
The length L and width w (in yd) of a rectangular soccer field should satisfy the inequalities 110 ≤ L ≤120 and 70 ≤ w ≤ 80. Express the possible diagonal lengths d (to the nearest
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y > 1 – x sin2xy < 5 – x²
The elements of an electric circuit dissipate p watts of power. The power pR dissipated by a resistor in the circuit is given by pR = Ri2, where R is the resistance (in Ω) and i is the current (in
A rectangular PV (photovoltaic) solar panel is designed to be 1.42 m long and supply 130 W/m2 of power. What must the width of the panel be in order to supply between 100 W and 150 W?
A beam is supported at each end, as shown in Fig. 17.16. Analyzing the forces leads to the equation F1 = 13 − 3d. For what values of d is F1 more than 6 N?Fig. 17.16. F 12 N -d- 2 N 4 m F 2
A breakfast cereal company guarantees the calorie count shown for each serving is accurate within 5%. If the package shows a serving has 200 cal, write an inequality for the possible calorie counts.
Use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.y > |x – 1|y < 4 + Inx
Determine the values of x for which the given radicals represent real numbers. √4-x
The object distance p (in cm) and image distance q (in cm) for a camera of focal length 3.00 cm is given by p = 3.00q/(q − 3.00). For what values of q is p > 12.0 cm?
The cross-sectional area A (in m2) of a certain trapezoidal culvert in terms of its depth d (in m) is A = 2d + d2. Graph the possible values of d and A if A is between 1m2 and 2m2.
Determine the values of x for which the given radicals represent real numbers. √x + 5
An electron microscope can magnify an object from 2000 times to 1,000,000 times. Assuming these values are exact, express these magnifications M as an inequality and graph them.
The total capacitance C of capacitors C1 and C2 in series is C−1 = C1−1 + C2−1 . If C2 = 4.00 μF, find C1 if C > 1.00 μF.
One pump can remove wastewater at the rate of 75 gal/min, and a second pump works at the rate of 45 gal/min. Graph the possible values of the time (in min) that each of these pumps operates such that
The mass m (in g) of silver plate on a dish is increased by electroplating. The mass of silver on the plate is given by m = 125 + 15.0t, where t is the time (in h) of electroplating. For what values
A busy person glances at a digital clock that shows 9:36. Another glance a short time later shows the clock at 9:44. Express the amount of time t (in min) that could have elapsed between glances by
During a given rush hour, the numbers of vehicles shown in Fig. 17.17 go in the indicated directions in a one-way-street section of a city. By finding the possible values of x and the equation
Determine the values of x for which the given radicals represent real numbers. x² + 3x
A rectangular field is to be enclosed by a fence and divided down the middle by another fence. The middle fence costs $4/ft and the other fence cost $8/ft. If the area of the field is to be 8000 ft2,
Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated inequality or system of inequalities.A rectangular computer chip is being designed such
For a ground temperature of T0 (in °C), the temperature T (in° C) at a height h (in m) above the ground is given approximately by T = T0 − 0.010h. If the ground temperature is 25°C, for
An Earth satellite put into orbit near Earth’s surface will have an elliptic orbit if its velocity v is between 18,000 mi/h and 25,000 mi/h. Write this as an inequality and graph these values of v.
Determine the values of x for which the given radicals represent real numbers. x-1 V2x + 5
The weight w (in N) of an object h meters above the surface of Earth is w = r2w (r + h)2, where r is the radius of Earth and w0 is the weight of the object at sea level. Given that r = 6380 km, if an
Find the indicated maximum and minimum values by the method of linear programming. The constraints are shown below the objective function.Maximum P: P = 2x +9y x ≥ 0, y ≥ 0 x + 4y < 13 3y - x ≤
Fossils found in Jurassic rocks indicate that dinosaurs flourished during the Jurassic geological period, 140 MY (million years ago) to 200 MY. Write this as an inequality, with t representing past
The value V after two years of an amount A invested in a mutual fund at an annual yield rate r is V = A(1 + r)2. If $10,000 is invested in order that the value V is between $11,000 and $11,500, what
The minimum legal speed on a certain interstate highway is 45 mi/h, and the maximum legal speed is 65 mi/h. What legal distances can a motorist travel in 4 h on this highway without stopping?
A Blu-ray disc spins at 1530 r/min when the innermost edge is being read by the laser, and gradually slows to a rate of 630 r/min at the outer edge. Use an inequality to express the angular velocity
A triangular postage stamp is being designed such that the height h is 1.0 cm more than the base b. Find the possible height h such that the area of the stamp is at least 3.0 cm2.
Find the indicated maximum and minimum values by the method of linear programming. The constraints are shown below the objective function.Maximum P: P = x + 2y x ≥ 1, y ≥ 0 3x + y ≤ 6 2x + 3y
The route of a rapid transit train is 40 km long, and the train makes five stops of equal length. If the train is actually moving for 1 h and each stop must be at least 2 min, what are the lengths of
A laser source is 2.0 in. from the nearest point P on a flat mirror, and the laser beam is directed at a point Q that is on the mirror and is x in. from P. The beam is then reflected to the receiver,
Find the indicated maximum and minimum values by the method of linear programming. The constraints are shown below the objective function.Minimum C: C = 3x + 4y x ≥ 0, y ≥ 1 > 2x + 3y ≥ 6 4x +
The velocity v of an ultrasound wave in soft human tissue may be represented as 1550 ± 60 m/s, where the ±60 m/s gives the possible variation in the velocity. Express the possible velocities by an
Find the indicated maximum and minimum values by the method of linear programming. The constraints are shown below the objective function.Minimum C: C = 2x + 4y x ≥ 0, y ≥ 0 x + 3y ≥ 6 4x + 7y
If the current from the source in Example 12 is i = 5 cos4πt and the diode allows only negative current to flow, write the inequalities and draw the graph for the current in the circuit as a
An oil company plans to install eight storage tanks, each with a capacity of x liters, and five additional tanks, each with a capacity of y liters, such that the total capacity of all tanks is
An open box (no top) is formed from a piece of cardboard 8.00 in. square by cutting equal squares from the corners, turning up the resulting sides, and taping the edges together. Find the edges of
A driver using the Google Maps app finds that it is 300 mi to her destination. If her speed always stays between 50 mi/h and 60 mi/h, use an inequality to express the required time t for the trip.
A plane takes off from Winnipeg and flies due east at 620 km/h. At the same time, a second plane takes off from the surface of Lake Winnipeg 310 km due north of Winnipeg and flies due north at 560
Under what conditions is |a + b| < |a| + |b|?
Is |a − b| < |a| + |b| always true? Explain.
For what values of x is the graph of y = x2 + 3x above the graph of y = 2x + 6?
Solve for x: a + |bx| < c given that a − c < 0.
Form an inequality of the form ax2 + bx + c < 0 with a > 0 for which the solution is −3 < x < 5.
If two adjacent sides of a square displayed on a TV screen expand 6.0 cm and 10.0 cm, respectively, how long is each side of the original square if the perimeter of the resulting rectangle is at
By means of an inequality, define the region above the line x − 3y − 6 = 0.
Draw a graph of the system y < 1 − x2 and y = x2.
Find a system of inequalities that would describe the region within the quadrilateral with vertices (0, 0), (4, 4), (0,−3), and (4,−3).
Find the values for which f(x) = (x − 2)(x − 3) is positive, zero, and negative. Use this information along with f(0) and f(5) to make a rough sketch of the graph of f(x).
Follow the same instructions as in Exercise 71 for the function f(x) = (x − 2)/(x − 3).Data from Exercises 71Find the values for which f(x) = (x − 2)(x − 3) is positive, zero, and negative.
Describe the region that satisfies the system y ≥ 2x − 3, y < 2x − 3.
Describe the region defined by y ≥ 2x − 3 or y < 2x − 3.
The value V (in $) of each building lot in a development is estimated as V = 65,000 + 5000t, where t is the time in years from now. For how long is the value of each lot no more than $90,000?
The cost C of producing two of one type of calculator and five of a second type is $50. If the cost of producing each of the second type is between $5 and $8, what are the possible costs of producing
City A is 600 km from city B. One car starts from A for B 1 h before a second car. The first car averages 60 km/h, and the second car averages 80 km/h for the trip. For what times after the first car
The pressure p (in kPa) at a depth d (in m) in the ocean is given by p = 101 + 10.1d. For what values of d is p > 500 kPa ?
Solve the given problems using inequalities. After conducting tests, it was determined that the stopping distance x (in ft) of a car traveling 60 mi/h was |x − 290| ≤ 35. Express this inequality
A heating unit with 80% efficiency and a second unit with 90% efficiency deliver 360,000 Btu of heat to an office complex. If the first unit consumes an amount of fuel that contains no more than
A rectangular parking lot is to have a perimeter of 180 m and an area of at least 2000 m2. What are the possible dimensions of the lot?
The electric power p (in W) dissipated in a resistor is given by p = Ri2, where R is the resistance (in Ω) and i is the current (in A). For a given resistor, R = 12.0 Ω, and the power varies
The reciprocal of the total resistance of two electric resistances in parallel equals the sum of the reciprocals of the resistances. If a 2.0-Ω resistance is in parallel with a resistance R, with a
The efficiency e (in %) of a certain gasoline engine is given by e = 100(1 − r−0.4), where r is the compression ratio for the engine. For what values of r is e > 50%?
A rocket is fired such that its height h (in mi) is given by h = 41t − t2. For what values of t (in min) is the height greater than 400 mi?
In developing a new product, a company estimates that it will take no more than 1200 min of computer time for research and no more than 1000 min of computer time for development. Graph the possible
In Example 6, change “inversely as x” to “inversely as the square of x” and then solve the resulting problem.Data from Example 6If y varies inversely as x, and x = 15 when y = 4, find the
A natural gas supplier has a maximum of 120 worker-hours per week for delivery and for customer service. Graph the possible combinations of times available for these two services.
A company produces two types of cell phones, the regular model and the deluxe model. For each regular model produced, there is a profit of $8, and for each deluxe model the profit is $15. The same
A company that manufactures stylus pens gets two different parts, A and B, from two different suppliers. Each package of parts from the first supplier costs $2.00 and contains 6 of each type of part.
In planning a new city development, an engineer uses a rectangular coordinate system to locate points within the development. A park in the shape of a quadrilateral has corners at (0, 0), (0, 20),
In Example 7, change 1.14 N to 1.35 N and then solve the resulting problem.Data from Example 7The frequency f of vibration of a wire varies directly as the square root of the tension T of the wire.
In Example 1(a), change radius r to diameter d, write the appropriate equation, and find the value of the constant of proportionality.Data from Example 1(a)The circumference c of a circle is
Express the ratio of 180 s to 4 min in simplest form.
In Example 6, change 5 parts tin to 7 parts tin.Data from Example 6An alloy is 5 parts tin and 3 parts lead. How many grams of each are in 40 g of the alloy? First, let x = the number of grams of tin
In Example 2, change 18 ft to 16 ft.Data from Example 2The length of a certain room is 24 ft, and the width of the room is 18 ft. Therefore, the ratio of the length to the width is 24/18 or 4/3.If
Determine each of the following as being either true or false. If it is false, explain why.The ratio of 25 cm to 50 mm is 5.

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