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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
In testing a new electric engine, an automobile company randomly selected 20 cars of a certain model and recorded the range (in mi) that the car could travel before the batteries needed recharging. The results are shown below.Draw a frequency histogram using the classes given in Exercise 15.Data
It has been previously established that for a certain type of AA battery (when newly produced), the voltages are distributed normally with μ = 1.50 V and σ = 0.05 V.What percent of the batteries have voltages above 1.64 V?
Find and interpret the values of r and r2 for the given data.Exercise 13Data from Exercises 13In an experiment on the photoelectric effect, the frequency of light being used was measured as well as the stopping potential (the voltage just sufficient to stop the photoelectric effect) with the
Determine the mode of the numbers of the given set.Set DD: 105, 108, 103, 108, 106, 104, 109, 104, 110, 108, 108, 104, 113, 106, 107, 106, 107, 109, 105, 111, 109, 108
Find the standard deviation for the indicated sets of numbers. A calculator may be used.The X-ray dosages in Exercise 25Data from Exercises 25The dosage (in mR) given by a particular X-ray machine was measured 20 times, with the following readings:Using the class limits 3.80, 3.90, 4.00, . . . ,
In testing a new electric engine, an automobile company randomly selected 20 cars of a certain model and recorded the range (in mi) that the car could travel before the batteries needed recharging. The results are shown below.Draw a relative frequency histogram using the classes given in Exercise
Find the standard deviation for the indicated sets of numbers. A calculator may be used.The alcohol percentages in Exercise Data from Exercises 27The blood alcohol level (in %) in the bloodstreams of those charged with DUI at a police check point were as follows:Using the class limits 0.06,
What percent of the tires will last between 85,000 km and 100,000 km?The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km.
A manufacturer of drone propellers randomly selects 1000 propellers from the production line each week and inspects them for defects. The number of defective propellers, along with the proportion of defective propellers, is given in the following table for a 20-week period:Plot the p chart. Week
Find the mean.In testing the water supply of a town, the volume V (in mL/L) of settleable solids over a 12-day period were as follows (readings of V < 0.15 mL/L are considered acceptable): 0.11, 0.15, 0.16, 0.13, 0.14, 0.13, 0.12, 0.14, 0.15, 0.13, 0.13, 0.12.
In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:Make a stem-and-leaf plot of these data (without split stems). 112, 91, 101, 85, 76, 115, 93, 126, 78, 86, 105, 107, 58, 86, 109, 111, 103, 105, 97,
What percent of the tires will last between 95,000 km and 115,000 km?The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km.
In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:Make a stem-and-leaf plot of these data using split stems. 112, 91, 101, 85, 76, 115, 93, 126, 78, 86, 105, 107, 58, 86, 109, 111, 103, 105, 97, 110,
The maker of electric fuses checks 500 fuses each day for defects. The number of defective fuses, along with the proportion of defective fuses for 24 days, is shown in the following table.For the p chart, find the values for the central line, UCL, and LCL. Day Number
Find the median.In testing the water supply of a town, the volume V (in mL/L) of settleable solids over a 12-day period were as follows (readings of V < 0.15 mL/L are considered acceptable): 0.11, 0.15, 0.16, 0.13, 0.14, 0.13, 0.12, 0.14, 0.15, 0.13, 0.13, 0.12.Find the mean.In testing the water
The maker of electric fuses checks 500 fuses each day for defects. The number of defective fuses, along with the proportion of defective fuses for 24 days, is shown in the following table.Plot the p chart. Day Number
In a sample of 5000 of these tires, how many can be expected to last more than 118,000 km?The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km.
Find the standard deviation for the indicated sets of numbers. A calculator may be used.The electrical usages in Exercise 31Data from Exercises 31In a particular month, the electrical usages, rounded to the nearest 100 kW · h (kilowatt-hours), of 1000 homes in a certain city were summarized as
Find the standard deviation for the indicated sets of numbers. A calculator may be used.The diameters in Exercise 33Data from ExercisesThe diameters of a sample of fiber-optic cables were measured (to the nearest 0.0001 mm) with the following results.Find the mean of the diameters. Diameter
In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:Make a frequency distribution table using the class limits 50, 60, 70, . . . , 130. 112, 91, 101, 85, 76, 115, 93, 126, 78, 86, 105, 107, 58, 86,
Find the standard deviation.In testing the water supply of a town, the volume V (in mL/L) of settleable solids over a 12-day period were as follows (readings of V < 0.15 mL/L are considered acceptable): 0.11, 0.15, 0.16, 0.13, 0.14, 0.13, 0.12, 0.14, 0.15, 0.13, 0.13, 0.12.
If the manufacturer guarantees to replace all tires that do not last 75,000 km, what percent of the tires may have to be replaced under this guarantee?The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard
In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:Find the relative frequencies using the class limits given in Exercise 21.Data from Exercises 21Make a frequency distribution table using the class
A sample of wind generators was tested for power output when the wind speed was 30 km/h. The following table gives the powers produced (to the nearest 10 W) and the number of generators for each power value.Find the median. Power (W) No. Generators Power (W) No. Generators 650 660 670 680 690 3 2
In testing the water supply of a town, the volume V (in mL/L) of settleable solids over a 12-day period were as follows (readings of V < 0.15 mL/L are considered acceptable): 0.11, 0.15, 0.16, 0.13, 0.14, 0.13, 0.12, 0.14, 0.15, 0.13, 0.13, 0.12.Draw a histogram.
In a sample of 100 of these tires, find the likelihood that the mean lifetime for the sample is less than 98,200.The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km.
A sample of wind generators was tested for power output when the wind speed was 30 km/h. The following table gives the powers produced (to the nearest 10 W) and the number of generators for each power value.Find the mean. Power (W) No. Generators Power (W) No. Generators 650 660 670 680 690 3 2
In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:Make a frequency histogram using the table in Exercise 21. Describe the shape.Data from Exercises 21Make a frequency distribution table using the class
A sample of wind generators was tested for power output when the wind speed was 30 km/h. The following table gives the powers produced (to the nearest 10 W) and the number of generators for each power value.Find the standard deviation. Power (W) No. Generators Power (W) No. Generators 650 660 670
Find the indicated measures of central tendency.Mode of the number of apps in Exercise 19Data from ExercisesUse the following data. In a random sample, 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below:112, 91, 101, 85,
Find the indicated measures of central tendency.Mean of X-ray dosages in Exercise 25Data from Exercises 25 Use the following data. The dosage (in mR) given by a particular X-ray machine was measured 20 times, with the following readings:4.25, 4.36, 3.96, 4.21, 4.44, 3.83, 4.37, 4.27, 4.33,
The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km. Answer the following question.What percent of the samples of 100 of these tires should have a mean lifetime of more than 102,000 km?
The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of 100,000 km and a standard deviation of 10,000 km. Answer the following question.If 144 of the tires are randomly selected, find the percent chance that the mean lifetime is more
Solve the systems of equations by determinants. As indicated, use the systems from Exercises.Exercises 38Data from Exercises 38Solve the given systems of equations using the inverse of the coefficient matrix. 3x + 2y + z = 22x + 3y − 6z = 3x + 3y + 3z = 1
Express each as a sum, difference, or multiple of logarithms. See Example 2.log4 7√xData from Example 2(a) Using Eq. (13.7), we may express log4 15 as a sum of logarithms:(b) Using Eq. (13.8), we may express log4 (5/3) as the difference of logarithms:(c) Using Eq. (13.9), we may express log4 (t2)
Express each as a sum, difference, or multiple of logarithms. See Example 2.Data from Example 2(a) Using Eq. (13.7), we may express log4 15 as a sum of logarithms:(b) Using Eq. (13.8), we may express log4 (5/3) as the difference of logarithms:(c) Using Eq. (13.9), we may express log4 (t2) as twice
Express each as a sum, difference, or multiple of logarithms. See Example 2.Data from Example 2(a) Using Eq. (13.7), we may express log4 15 as a sum of logarithms:(b) Using Eq. (13.8), we may express log4 (5/3) as the difference of logarithms:(c) Using Eq. (13.9), we may express log4 (t2) as twice
Solveby first writing it in quadratic form as shown in Section 14.3. √x - 2 = = √x 2 + 12
Solve the given equation. √3x - 2√x + 7 = 1
Solve the given systems of equations by use of a calculator.y = x² − 2xy=1− e−x + x
Assume Earth is a sphere, with x2 + y2 = 41 as the equation of a circumference (distance in thousands of km). If a meteorite approaching Earth has a path described as y2 = 20x + 140, will the meteorite strike Earth? If so, where?
In a marketing survey, a company found that the total gross income for selling t tables at a price of p dollars each was $35,000. It then increased the price of each table by $100 and found that the total income was only $27,000 because 40 fewer tables were sold. Find p and t.
Find the value of x. (In each, the expression on the right is called a continued radical. Also, . . . means that the pattern continues indefinitely.) X = √6-√√6-√6-
Find the value of x. (In each, the expression on the right is called a continued radical. Also, . . . means that the pattern continues indefinitely.) x = √√2 + √√2 + √√2+...
If the value of a home increases from v1 to v2 over n years, the average annual rate of growth (as a decimal) is given bySuppose the value of a home increases on average by 3.6% per year over 10 years. If its value at the end of the 10-year period is $325,000, find its value at the beginning of the
The speed s (in m/s) at which a tsunami wave moves is related to the depth d (in m) of the ocean according to s = √gd, where g is the acceleration of gravity (9.8 m/s2). If a wave from the 2004 Indian Ocean tsunami was traveling at 195 m/s, estimate the depth of the ocean at that point.
In the theory dealing with a suspended cable, the equationis used. Solve for m. y = √√s² m² m 2 -
A wrench is dropped by a worker at a construction site. Four seconds later the worker hears it hit the ground below. How high is the worker above the ground? (The velocity of sound is 1100 ft/s, and the distance the wrench falls as a function of time is s = 16t 2.)
The rectangular screen for a laptop computer has an area of 840 cm2 and a perimeter of 119 cm. What are the dimensions of the screen?
Solve the given equations algebraically and check the solutions with a calculator. 2√3x + 1 √√√x-1= 6
A freighter is 5.2 km farther from a Coast Guard station on a straight coast than from the closest point A on the coast. If the station is 8.3 km from A, how far is it from the freighter?
A point D on Denmark’s largest island is 2.4 mi from the nearest point S on the coast of Sweden (assume the coast is straight, which is nearly the case). A person in a motorboat travels straight from D to a point on the beach x mi from S and then travels x mi farther along the beach away from S.
Solve the given equations algebraically and check the solutions with a calculator. 9 = L + x + L + zx^
Solve the given equations algebraically and check the solutions with a calculator. 3√x + √√√x-9 = 11
The length of the roller belt in Fig. 14.24 is 28.0 ft. Find x. x Fig. 14.24 3.0 ft 3.0 ft X
Solve the given equations algebraically and check the solutions with a calculator. √√x³7= x - 1
Solve the given equations algebraically and check the solutions with a calculator.(x + 1)4 − 54 = 3(x + 1)2
The velocity v of an object that falls through a distance h is given by v = √2gh, where g is the acceleration due to gravity. Two objects are dropped from heights that differ by 10.0 m such that the sum of their velocities when they strike the ground is 20.0 m/s. Find the heights from which they
Solvefor x. Check using the graph on a calculator. √√√√√x-1=2
Ifand f(x + 1) = 2, find x. f(x) = √8 √8 - 2x
Doctors can estimate the surface area A of an adult body (in m2) using the equationwhere H is height (in cm) and W is weight (in kg). Estimate the height of a person who has weight of 81.2 kg and a surface area of 1.25 m2. A = HW 3600'
Solve for x and y: x2 − y2 = 2a + 1; x − y = 1
Solve for x: log(√x + 38) − log x = 1
The frequency ω of a certain RLC circuit is given bySolve for C. W = R² + 4(L/C) + R 2L
If two objects collide and the kinetic energy remains constant, the collision is termed perfectly elastic. Under these conditions, if an object of mass m1 and initial velocity u1 strikes a second object (initially at rest) of mass m2 , such that the velocities after collision are v1 and v2 ,
Use a graphing calculator to solve the following system of three equations: x2 + y2 = 13, y = x − 1, xy = 6.
Algebraically solve the following system of three equations: y = −x2, y = x − 1, xy = 1. Explain the results.
The equation V = e2cr−2 − e2Zr−1 is used in spectroscopy. Solve for r.
In an experiment, an object is allowed to fall, stop, and then fall for twice the initial time. The total distance the object falls is 45 ft. The equations relating the times t1 and t2 (in s) of fall are 16t12 + 16t22 = 45 and t2 = 2t1 Find the times of fall.
A rectangular field is enclosed by fencing and a wall along one side and half of an adjacent side. See Fig. 14.25. If the area of the field is 9000 ft2 and 240 ft of fencing are used, what are the dimensions of the field? Fig. 14.25 Wall A = 9000 ft² 240 ft of fencing Fencing
A circuit on a computer chip is designed to be within the area shown in Fig. 14.26. If this part of the chip has an area of 9.0 mm2 and a perimeter of 16 mm, find x and y.Fig. 14.26. 2y X 2x y
The perimeter of a banner in the shape of an isosceles triangle is 72 in., and its area is 240 in.2. Using a calculator, graphically find the lengths of the sides of the banner.
A trough is made from a piece of sheet metal 12.0 in. wide. The cross section of the trough is shown in Fig. 14.28. Find x. Fig. 14.28 X X Vx+ 20 12.0 in. X
The circular solar cell and square solar cell shown in Fig. 14.29 have a combined surface area of 40.0 cm2. Find the radius of the circular cell and the side of the square cell. Fig. 14.29 -7.00 cm
For the plywood piece shown in Fig. 14.27, find x and y. X 7.50 ft X Fig. 14.27 4.00 ft
A plastic band 19.0 cm long is bent into the shape of a triangle with sidesand 9. Find x. √x 1, √5x - 1, –
The viewing window on a graphing calculator has an area of 1770 mm2 and a diagonal of 62 mm. What are the length and width of the rectangle?
In Example 3, change the x + 4 to x + 3 and then find the remainder.Data from Example 3By using the remainder theorem, determine the remainder when 3x3 − x2 − 20x + 5 is divided by x + 4.In using the remainder theorem, we determine the remainder when the function is divided by x − r by
Using a computer, an engineer designs a triangular support structure with sides (in m) of and 4.00 m. If the perimeter is to be 9.00 m, the equation to be solved isThis equation can be solved by either of two methods used in this chapter. Write one or two paragraphs identifying the methods and
In Example 2, change the middle term of the second factor to 2x.Data from Example 2Consider the equation f(x) = (x − 1)3 (x2 + x + 1) = 0.The factor (x − 1)3 shows that there is a triple root of 1, and there is a total of five roots, because the highest-power term would be x5 if we were to
In Example 6, change the middle three terms to the left of the = sign to −3x3 + 7x2 + 7x, given the same root.Data from Example 6Solve the equation 2x4 − 5x3 + 11x2 − 3x − 5 = 0, given that 1 + 2 j is a root. Because 1 + 2j is a root, we know that 1 − 2j is also a root. Using synthetic
A Coast Guard ship travels from Houston to Mobile, and later it returns to Houston at a speed that is 6.0 mi/h faster. If Houston is 510 mi from Mobile and the total travel time is 35 h, find the speed of the ship in each direction.
Two trains are approaching the same crossing on tracks that are at right angles to each other. Each is traveling at 60.0 km/h. If one is 6.00 km from the crossing when the other is 3.00 km from it, how much later will they be 4.00 km apart (on a direct line)?
In Example 4, change all signs between terms. (Leave the first term positive.)Data from Example 4For the equation 4x5 − x4 − 4x3 + x2 − 5x − 6 = 0, we writeThus, there are no more than three positive and two negative roots. f(x) = = 4x5x4 - 4x³ + x² - 5x - 6 f(-x) = -4x5x4 + 4x³ +
In Example 5, change the + sign before the 5x to −.Data from Example 5Find the roots of the equation 2x3 + x2 + 5x − 3 = 0. Because n = 3, there are three roots. If we can find one of these roots, we can use the quadratic formula to find the other two. We havewhich shows there is one positive
Determine each of the following as being either true or false. If it is false, explain why.If 3x2 + 5x − 8 is divided by x − 2, the remainder is 12.
Is −3 a zero for the function 2x3 + 3x2 + 7x − 6? Explain.
Determine each of the following as being either true or false. If it is false, explain why.Using synthetic division to divide 2x3 − 3x2 − 23x + 12 by x + 3, the bottom row of numbers is 2 −9 −4 0.
Find the remaining roots of the equation x4 − 2x3 − 7x2 + 20x − 12 = 0; 2 is a double root.
In Example 4(a), change the t + 1 to t − 1 and then determine if t − 1 is a factor.Data from Example 4(a)We determine that t + 1 is a factor of f(t) = t3 + 2t2 − 5t − 6 because f(−1) = 0, as we now show: f(−1) = (−1)3 + 2(−1)2 − 5(−1) − 6 = −1 + 2 + 5 − 6 = 0.
Find the inverse of each of the given matrices by transforming the identity matrix, as in Examples 2–4.Data from Example 4Find the inverse of the matrixTherefore, the required inverse matrix iswhich may be checked by multiplication. See Fig. 16.8 for a calculator window showing A−1 in decimal
Find the roots of the given equations by inspection.(x + 3)(x2 − 4) = 0
Determine each of the following as being either true or false. If it is false, explain why.Without solving, it can be determined that 1/8 is a possible rational root of the equation 4x4 − 3x3 + 5x2 − x + 8 = 0.
Use matrices A and B.Find A2 , A3, and A4. 10 4-[18] A 3 B 0 1 0 001 100
Use matrices A and B.Show that (A2)2 = A4 . 10 4-[18] A 3 B 0 1 0 001 100
Use matrices A and B.Show that B3 = I. 10 4-[18] A 3 B 0 1 0 001 100
Solve the given systems of equations by using the inverse of the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check. See Example 4.3x − y = 47x + 2y = 18Data from Example 4Use a calculator to perform the necessary matrix
Make the indicated changes in the determinant at the right, and then solve the indicated problem. Assume the elements are nonzero, unless otherwise specified.Evaluate the determinant if b = c = f = 0. a b C de f h i 00 8
Find the indicated matrices using a calculator.1/3B −1/2A A = c = C 6-3 4 -5 B = -1 4 -7 2 -6 11 3 -9 D = 12 -6 79-6 -4 0 8
Find the inverses of the given matrices. Check each by using a calculator. 3 1-4 -3 1-2 -6 0 3
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