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mathematics
basic technical mathematics

Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions

Solve the given systems of equations by any appropriate method.A 20-ft crane arm with a supporting cable and with a 100-lb box suspended from its end has forces acting on it, as shown in Fig. 5.53. Find the forces (in lb) from the following equations. F₁ + 2.0F₂ = 280 0.87F, F30 3.0F₁ -
Determine the value of k that makes the system inconsistent. kx - 2y = 5 4x+6y= 1
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A computer analysis showed that the temperature T of the ocean water within 1000 m of a nuclear-plant discharge pipe was given bywhere x is the distance from the pipe and a
Solve the given systems of equations by any appropriate method.In applying Kirchhoff’s laws to the electric circuit shown in Fig. 5.54, the following equations result. Find the indicated currents (in A). i₁ + 1₂ + 13 = 0 5.201₁ - 3.25i₂ 3.25i₂ - 2.62i3 8.33 6.45 = 6.45 9.80 = -
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.As a 40-ft pulley belt makes one revolution, one of the two pulley wheels makes one more revolution than the other. Another wheel of half the radius replaces the smaller
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.In measuring the angles of a triangular parcel, a surveyor noted that one of the angles equaled the sum of the other two angles. What conclusion can be drawn?
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A total of 42.0 tons of crushed shale-oil rock is to be refined to extract the oil. The first contains 18.0 gal/ton and the second contains 30.0 gal/ton. How much of each
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.An online merchandise company charges $4 for shipping orders of less than $50, $6 for orders from $50 to $200, and $8 for orders over $200. One day the total shipping charges
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.The sales representatives of a company have a choice of being paid 10%, of their sales in a month, or $2400 plus 4%, of their sales in the month. For what monthly sales is
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A study found that the fuel consumption for transportation contributes a percent p1 that is 16% less than the percent p2 of all other sources combined. Find p1 and p2.
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A certain amount of a fuel contains 150,000 Btu of potential heat. Part is burned at 80% efficiency, and the rest is burned at 70%, efficiency, such that the total amount of
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A person invested a total of $20,900 into two mutual funds. The yields of the funds after one year were 6.00% and 5.00%. If the total income from both funds was $1170, how
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A total of 42 tons of two types of ore is to be loaded into a smelter. The first type contains 6.0% copper, and the second contains 2.4% copper. Find the necessary amounts of
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A satellite is to be launched from a space shuttle. It is calculated that the satellite’s speed will be 24,200 km/h if launched directly ahead of the shuttle or 21,400 km/h
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.For the triangular truss shown in Fig. 5.59, ∠A is 55° less than twice ∠B, and ∠C is 25° less than ∠B. Find the measures of the three angles. B A Fig. 5.59 C
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.Twelve equal rectangular ceiling panels are placed as shown in Fig. 5.57. If each panel is 6.0 in. longer than it is wide and a total of 132.0 ft of edge and middle strips is
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.In an office building one type of office has 800 ft2 and rents for $900/month. A second type of office has 1100 ft2 and rents for $1250/month. How many of each are there if
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.The weight of a lever may be considered to be at its center. A 20-ft lever of weight w is balanced on a fulcrum 8 ft from one end by a load L at that end. If a load of 4L is
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A study showed that the percent of persons age 25 and older with a bachelor's degree was 29.9% in 2010 and 36.4% in 2020. Assuming the increase to be linear, find an equation
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.The velocity v of sound is a function of the temperature T according to the function v = aT + b, where a and b are constants. If v = 337.5 m/s for T = 10.0°C and v = 346.6
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.The power (in W) dissipated in an electric resistance (in Ω) equals the resistance times the square of the current (in A). If1.0 A flows through resistance R1 and 3.0 A
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.Two fuel mixtures, one of 2.0% oil and 98.0% gasoline and another of 8.0% oil and 92.0% gasoline, are to be used to make 10.0 L of a fuel that is 4.0% oil and 96.0% gasoline
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A manufacturer produces three models of smart watches in a year. Four times as many of model A are produced as model C, and 7000 more of model B than model C. If the total
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.Three computer programs A, B, and C require a total of 11 GB (gigabytes) of hard-disk memory. A requires double the memory of C, and C requires 1 GB more than B. How much
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.In a laboratory, electrolysis was used on a solution of sulfuric acid, silver nitrate, and cupric sulfate, releasing hydrogen gas, silver, and copper. A total mass of 1.750 g
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.Gold loses about 5.3%, and silver about 10%, of its weight when immersed in water. If a gold-silver alloy weighs 6.0 N in air and 5.6 N in water, find the weight in air of
Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits.A person engaged a workman for 48 days. For each day that he labored, he received 24 cents, and for each day he was idle, he paid 12 cents for his board. At the end of the 48
A measure of the light beam that can be passed through an optic fiber is its numerical aperture N. For a particular optic fiber, N is a function of the index of refraction n of the glass in the fiber, given byPlot the graph for n ≤ 2.00. N√n² 1.69. -
Solve the indicated equations graphically. Assume all data are accurate to two significant digits unless greater accuracy is given.A computer model shows that the cost (in dollars) to remove x percent of a pollutant from a lake isWhat percent can be removed for $25,000? C = 8000x 100 - x
The number of times S that a certain computer can perform a computation faster with a multiprocessor than with a uniprocessor is given bywhere n is the number of processors. Plot S as a function of n. S 5n 4+n
F = x4 − 12x3 + 46x2 − 60x + 25 is the force (in N) exerted by a cam on the arm of a robot, as shown in Fig. 3.25. Noting that x varies from 1 cm to 5 cm, plot the graph.Fig. 3.25. 1 cm 5 cm
Sketch the graph of a function for which the domain is all values of x and the range is −2 < y < 2.
The concentration C (in mg/L) of a drug in a patient’s bloodstream t hours after taking a pill is given byThe patient should receive a second dose after the concentration has dropped below 1.50 mg/L. How long will this take? C = 10.0t t² + 1.00
Solve the indicated equations graphically. Assume all data are accurate to two significant digits unless greater accuracy is given.A rectangular storage bin is to be made from a rectangular piece of sheet metal 12 in. by 10 in., by cutting out equal corners of side x and bending up the sides. See
If the function y = √x − 1 is shifted left 2 and up 1, what is the resulting function?
Solve the indicated equations graphically. Assume all data are accurate to two significant digits unless greater accuracy is given.In finding the illumination at a point x feet from one of two light sources that are 100 ft apart, it is necessary to solve the equation 9x3 − 2400x2 + 240,000x −
Explain how to show the graph of the relation y2 = x on a graphing calculator, and then display it on the calculator. See Example 8 on page 99.Data from Example 8Use the vertical line test to determine whether the relation y2 = x represents a function. By letting x = 0, 1, and 4, we get the table
If the function y = 3 − 2x is shifted right 1 and down 3, what is the resulting function?
In Fig. 3.26, part of the graph of y = 2x2 + 0.5 is shown. What is the area of the rectangle, if its vertex is on the curve, as shown?Fig. 3.26. 0 У y=f(x) 0.5 0.6 -Х
The voltage V across a capacitor in a certain electric circuit for a 2-s interval is V = 2t during the first second and V = 4 − 2t during the second second. Here, t is the time (in s). Plot V as a function of t.
For f (x) = 2x3 − 3, display the graphs of f(x) and f (−x) on a graphing calculator. Describe the graphs in relation to the y-axis
The cutting speed s (in ft/min) of a saw in cutting a particular type of metal piece is given by where t is the time in seconds. What is the maximum cutting speed in this operation (to two significant digits)? S = √t - 4t²,
Given that the point (1, 2) is on the graph of y = f(x), must it be true that f(2) = 1? Explain.
Referring to Exercise 60, explain how to determine the maximum possible capacity for a storage bin constructed in this way. What is the maximum possible capacity (to three significant digits)?Data from Exercises 60A rectangular storage bin is to be made from a rectangular piece of sheet metal 12
For f (x) = 2x3 − 3, display the graphs of f(x) and −f (x) on a graphing calculator. Describe the graphs in relation to the x-axis.
Is it possible that the points (2, 3), (5,−1), and (−1, 3) are all on the graph of the same function? Explain.
On a taxable income of x dollars, a certain city’s income tax T is defined as T = 0.02x if 0 < x ≤ 20, 000, T = 400 + 0.03(x − 20, 000) if x > 20,000. Graph T = f (x) for 0 ≤ x < 100,000.
For f (x) = x + 1 and is f (x) = g(x)? Explain. Display the graph of each on a calculator (being very careful when choosing the window settings for x). g(x) = 2 x² - 1 x-1
A hot-water faucet is turned on. (a) Sketch a reasonable graph of the water temperature as a function of time. (b) Compare to a typical situation described bywhere T is the water temperature (in °C) and t is the time (in s). T = t³ + 80 0.015³ +4
For the functions f (x) = |x| and g(x) = √x2 , use a graphing calculator to determine whether or not f (x) = g(x) for all real x.
An equation used in electronics with a transformer antenna isFor I = f (m), find f(0.55). I = 12.5√1 + 0.5m².
The temperature T (in °C) recorded on a day during which a cold front passed through a city was T = 2 + h for 6 < h < 14, T = 16 − 0.5h for 14 ≤ h < 20, where h is the number of hours past midnight. Graph T as a function of h for 6 < h < 20.
Plot graph of f(x) = 1 Å √x+1 (for x < 0) (for x > 0)
Plot the graphs of y = x and y = |x| on the same coordinate system. Explain why the graphs differ.
A balloon is being blown up at a constant rate. (a) Sketch a reasonable graph of the radius of the balloon as a function of time. (b) Compare to a typical situation that can be described by r = 3√3t, where r is the radius (in cm) and t is the time (in s).
Express the area of a circle as a function of the side s of a square inscribed within the circle (all four corners of the square are on the circle).
The percent p of wood lost in cutting it into boards 1.5 in. thick due to the thickness t (in in.) of the saw blade is Find p if t = 0.4 in. That is, since p = f (t), find f(0.4). P = 100t t + 1.5
Plot the graphs of (a) y = x + 2 (b) Explain the difference between the graphs. y = x² - 4 2 x-2
Plot the graphs of y = 2 − x and y = |2 − x| on the same coordinate system. Explain why the graphs differ.
The angle A (in degrees) of a robot arm with the horizontal as a function of time t (for 0.0 s to 6.0 s) is given by A = 8.0 + 12t2 − 2.0t3. What is the greatest value of A to the nearest 0.1°? See Fig. 3.43. A Fig. 3.43 C
Graph the indicated functions.Plot the graphs of (a) y = x2 − x + 1 (b) Explain the difference between the graphs. y x³ + 1 x + 1
The electric power it P(in W) produced by a certain battery iswhere R is the resistance (in Ω) in the circuit. What is the maximum power produced? P = 24R R² + 1.40R+ 0.49'
Display the graph of y = cx3 with c = −2 and with c = 2. Describe the effect of the value of c.
Determine whether or not the indicated graph is that of a function.Fig. 3.27(b) y 0 (b) x
Display the graph of y = cx4 with c = 4 and with c =1/4. Describe the effect of the value of c.
Determine whether or not the indicated graph is that of a function.Fig. 3.27(a) 0 у (a) X
Determine whether or not the indicated graph is that of a function.Fig. 3.27(c) y 0 (c) X
Determine whether or not the indicated graph is that of a function.Fig. 3.27(d) Х (P) У 0
Plot the graphs of the indicated functions.When El Niño, a Pacific Ocean current, moves east and warms the water off South America, weather patterns in many parts of the world change significantly. Special buoys along the equator in the Pacific Ocean send data via satellite to monitoring stations.
The change C (in in.) in the length of a 100-ft steel bridge girder from its length at 40°F as a function of the temperature T is given by C = 0.014(T − 40). Plot the graph for T = 0°F to T = 120°F.
The profit P (in dollars) a retailer makes in selling 50 game controllers is given by P = 50(p − 80), where p is the selling price. Plot P as a function of p for p = $70 to p = $100.
There are 500 L of oil in a tank that has the capacity of 100,000 L. It is filled at the rate of 7000 L/h. Determine the function relating the number of liters N and the time t while the tank is being filled. Plot N as a function of t.
A medical researcher exposed a virus culture to an experimental vaccine. It was observed that the number of live cells N in the culture as a function of the time t (in h) after exposure was given byPlot the graph of N = f(t). N = 1000 √t + 1*
The length L (in cm) of a pulley belt is 12 cm longer than the circumference of one of the pulley wheels. Express L as a function of the radius r of the wheel and plot L as a function of r.
To draw the approximate shape of an irregular shoreline, a surveyor measured the distances d from a straight wall to the shoreline at 20-ft intervals along the wall, as shown in the following table. Plot the graph of distance d as a function of the distance D along the wall. D (ft) d (ft) 15 32 0
The pressure loss P (in lb/in.2 per 100 ft) in a fire hose is given by P = 0.00021Q2 + 0.013Q, where Q is the rate of flow (in gal/min). Plot the graph of P as a function of Q.
A thermograph measures the infrared radiation from each small area of a person’s skin. Because the skin over a tumor radiates more than skin from nearby areas, a thermograph can help detect cancer cells. The emissivity e (in %) of radiation as a function of skin temperature T (in K) is e = f (T)
The vertical sag s (in ft) at the middle of an 800-ft power line as a function of the temperature T (in °F) is given in the following table. See Fig. 3.44. For the function s = f(T), find f(47) by linear interpolation. T (°F) s (ft) 0 20 40 10.2 10.6 11.1 Fig. 3.44 60 11.7 800 ft S
If an amount A is due on a certain credit card, the minimum payment p is A, if A is $20 or less, or $20 plus 10% of any amount over $20 that is owed. Find p = f(A), and graph this function
For a certain laser device, the laser output power P (in mW) is negligible if the drive current i is less than 80 mA. From 80 mA to 140 mA, P = 1.5 × 10−6i3 − 0.77. Plot the graph of P = f(i).
The percent p of a computer network that is in use during a particular loading cycle as a function of the time t (in s) is given in the following table. Plot the graph of p = f (t). t (s) P (%) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 45 85 90 85 85 60 10 1.6 0
It is determined that a good approximation for the cost C (in cents/mi) of operating a certain car at a constant speed v (in mi/h) is given by C = 0.025v2 − 1.4v + 35. Plot C as a function of v for v = 10 mi/h to v = 60 mi/h.
The electric field E (in V/m) from a certain electric charge is given by E = 25/r2 , where r is the distance (in m) from the charge. Plot the graph of E = f(r) for values of r up to 10 cm.
In an experiment measuring the pressure p (in kPa) at a given depth d(in m) of seawater, the results in the following table were found. Plot the graph of p = f (d) and from the graph determine f(10).
Answer the given questions about the indicated examples of this section.In Example 2, if (4, 3) replaces (3, 4), what are the values?Data from Example 2Find the values of the trigonometric functions of the standard-position angle θ with its terminal side passing through the point (3, 4).By placing
A person 250 mi from home starts toward home and travels at 60 mi/h for the first 2.0 h and then slows down to 40 mi/h for the rest of the trip. How long does it take the person to be 70 mi from home?
Make the given changes in the indicated examples of this section and then find the indicated values.In Example 2, in line 4, change 58.5° to 62.1° and then find the distance.Data from Example 2The Goodyear blimp (now, actually a semi-rigid airship) is 1850 ft above the ground and south of the
Make the given changes in the indicated examples of this section and then find the indicated values.In Example 2, interchange a and b and then find the values.Data from Example 2Given a = 4, b = 7, and c = √65 (see Fig. 4.33), find sin A, cos A, tan A, sin B, cos B, and tan B in exact form and in
One industrial cleaner contains 30% of a certain solvent, and another contains 10% of the solvent. To get a mixture containing 50 gal of the solvent, 120 gal of the first cleaner is used. How much of the second must be used?
The solubility s (in kg/m3 of water) of a certain type of fertilizer is given by s = 135 + 4.9T + 0.19T2 , where T is the temperature (in °C). Find T for s = 500 kg/m3.
Make the given changes in the indicated examples of this section and then find the indicated values.In Example 4, change cosθ to sinθ and then find the angle.Data from Example 4If cosθ = 0.3527, which means that θ = cos−1 0.3527 (θ is the angle whose cosine is 0.3527), we can use a
A 2.00-L (2000-cm3) metal container is to be made in the shape of a right circular cylinder. Express the total area A of metal necessary as a function of the radius r of the base. Then find A for r = 6.00 cm, 7.00 cm, and 8.00 cm.
In an oil pipeline, the velocity v (in ft/s) of the oil as a function of the distance x (in ft) from the wall of the pipe is given by v = 7.6x − 2.1x2. Find x for v = 5.6 ft/s. The diameter of the pipe is 3.50 ft.
Find the smallest positive angle and the smallest negative angle (numerically) coterminal with but not equal to −165°.
Determine each of the following as being either true or false.A standard position angle of 205° is a second-quadrant angle.
In Example 5, change sec 27.82° to csc 27.82° and then find the value.Data from Example 5To find the value of sec 27.82°, we use the fact thatsec 27.82° = (cos 27.82°)-1 = 1.131Either of the above two options can be evaluated on a calculator to obtain the result 1.131, which has been rounded
In Example 1, find another angle that is coterminal with the given angle.Data from Example 1Determine the measures of two angles that are coterminal with an angle of 145.6°.Because there are 360° in one complete rotation, we can find a coterminal angle by adding 360° to the given angle of
Make the given changes in the indicated examples of this section and then find the indicated values.In Example 3, in line 2, change 2.000 min to 3.000 min and then find the altitude.Data from Example 3A missile is launched at an angle of 26.55° with respect to the horizontal. If it travels in a
In Example 4, if 4/7 replaces 3/7, what are the values?Data from Example 4If we know that sin θ = 3/7, and that θ is a first-quadrant angle, we know the ratio of the ordinate to the radius vector (y to r) is 3 to 7. Therefore, the point on the terminal side for which y = 3 can be found by use of
Express 37°39´ in decimal form.

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