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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Express the given numbers in exponential form.4650∠326.5°
Perform the indicated operations, expressing all answers in the form a + bj.(7 − j) − (4 − 4j) − (j − 6)
An ac circuit contains the given combination of circuit elements from among a resistor (R = 45.0Ω), a capacitor (C = 86.2 μF), and an inductor (L = 42.9 mH). If the frequency in the circuit is f = 60.0 Hz, find (a) The magnitude of the impedance (b) The phase angle between the current
Express each number in terms of j.−√−0.01
An ac circuit contains the given combination of circuit elements from among a resistor (R = 45.0Ω), a capacitor (C = 86.2 μF), and an inductor (L = 42.9 mH). If the frequency in the circuit is f = 60.0 Hz, find (a) The magnitude of the impedance (b) The phase angle between the current
Use the circuit shown in Fig. 12.29. The current in the circuit is 5.75 mA. Determine the indicated quantities.(a) The magnitude of the impedance across the resistor, inductor, and capacitor (between points a and d).(b) The phase angle between the current and the voltage for the combination in
Express 3.47 − 2.81j in exponential form.
Locate the given numbers in the complex plane.−3j
Perform the indicated operations, expressing all answers in simplest rectangular form.(6 − 2j) + (4 + j)
Simplify, expressing the result in rectangular form: 4j 2 – 5 + 3j
Express the given numbers in exponential form.375.5[cos(−95.46°) + j sin(−95.46°)]
Express each number in terms of j.−√−4
Perform the indicated operations, expressing all answers in the form a + bj.(7j − 6) − (19 − 3j)
Perform the indicated operations. Leave the result in polar form.[3(cos120° + j sin120°)][5(cos45° + j sin 45°)]
Perform the indicated operations on the resulting expressions if the given changes are made in the indicated examples of this section.In Example 3(b), change the sign in the second denominator from + to − and then simplify.Data from Exercises 3(b) j + 2 3+j = = = (3 + j) + 2j j(3 +
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 6, replace 2j with −2j and then find the roots.Data from Example 6Find the two square roots of 2j.First, we write 2j in polar form as 2j = 2(cos
Locate the given numbers in the complex plane.10
Represent each complex number graphically and give the polar form of each.12j − 5
Use the circuit shown in Fig. 12.29. The current in the circuit is 5.75 mA. Determine the indicated quantities.(a) The magnitude of the impedance across the resistor and the inductor (between points a and c).(b) The phase angle between the current and the voltage for the combination in (a).(c) The
Determine each of the following as being either true or false. If it is false, explain why. The phase angle ∅ for an impedance Z = 4 − 4j is 45°.
Express the given numbers in exponential form.2.10(cos 588.7° + j sin 588.7°)
Express each number in terms of j.√−121
Perform the indicated operations, expressing all answers in the form a + bj.(−4 − j) + (−7 − 4j)
Use the circuit shown in Fig. 12.29. The current in the circuit is 5.75 mA. Determine the indicated quantities.The voltage across the resistor (between points a and b). a R = 2250 2 ww b XL = 1750 02 Fig. 12.29 C Xc=1400 HH d ac voltage source
Perform the indicated operations on the resulting expressions if the given changes are made in the indicated examples of this section.In Example 3(a), change the sign in the denominator from + to − and then simplify.Data from Example 3(a)This could be written in the form a + bj as 13/25 −
Locate the given numbers in the complex plane.−5 + j
Determine each of the following as being either true or false. If it is false, explain why.2eπj = −2
Represent each complex number graphically and give the polar form of each.−8 − 15j
In Example 6(c), change the 5 on the right side of the equation to −5 and then solve.Data from Example 6(c)What values of x and y satisfy the equation x + 3(xj + y) = 5 − j − jy ?Rewriting each side in the form a + bj, and equating real parts and equating imaginary parts, we have (x + 3y) +
Express the given numbers in exponential form.575(cos 135.0° + j sin 135°)
Express 2 − 7j in polar form.
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 4, change the exponent to 5 and then find the result.Data from Example 4Using DeMoivre’s theorem, find (2 + 3 j)3. Converting to polar form, 2 + 3j
The frequency of a certain electric circuit is given byExpress this in simplest rationalized radical form. 2π 1 LC₂C₂ 2 VC₁ + C₂ 1
In an experiment, a laser beam follows the path shown in Fig. 11.10. Express the length of the path in simplest radical form. 3 cm 3 cm 2 cm 1 cm
A square is decreasing in size on a computer screen. To find how fast the side of the square changes, we must rationalize the numerator ofPerform this operation. √A + h = √A h
The compression ratio r of a certain gasoline engine is related to the efficiency e (in %) of the engine by What compression ratio is necessary to have an efficiency of 55%? 100 - e 100 -2.5
The flow rate V (in ft/s) through a storm drain pipe is found from V = 110(0.55)2/3 (0.0018)1/2 . Find the value of V.
In analyzing the orbit of an Earth satellite, the expressionis used. Combine terms under the radical and express the result with the denominator rationalized. 2E h m GM 2
A plane takes off 200 mi east of its destination, and a 50 mi/h wind is from the south. If the plane’s velocity is 250 mi/h and its heading is always toward its destination, its path can be described as y = 100[(0.005x)0.8 − (0.005x)1.2], 0 ≤ x ≤ 200mi. Sketch the path and check using a
The expression (1 − v2/c2)−1/ 2 arises in the theory of relativity. Simplify, using only positive exponents in the result.
The value P of an object originally worth P0 and that appreciates at an annual rate of r for t years is given by P = P0(1 + r)t . Solve for r.
A square plastic sheet of side x is stretched by an amount equal to √x horizontally and vertically. Find the expression for the percent increase in the area of the sheet.
The speed v of a ship of weight W whose engines produce power P is v = k3√/P/W. Express this equation (a) With a fractional exponent (b) As a radical with the denominator rationalized.
Kepler’s third law of planetary motion may be given as T = kr3/2 , where T is the time for one revolution of a planet around the sun, r is its mean radius from the sun, and k = 1.115 × 10−12 year/mi3/2. Find the time for one revolution of Venus about the sun if r = 6.73 × 107mi.
The average annual increase i (in %) of the cost of living over n years is given by i = 100[(C2/C1)1/n − 1], where C1 is the cost of living index for the first year and C2 is the cost of living index for the last year of the period. Evaluate i if C1 = 237 and C2 = 258 are the values
Without a calculator, show that the given equations are true. Then verify each result with a calculator. √√√√3+1(√3 - 1) = √√2(√3 - 1)
Evaluate x if x−1/3 = 0.2.
Without a calculator, show that the given equations are true. Then verify each result with a calculator. √√√√2-1(√2+1) = √√√√√√2 + 1
Evaluate 3x2 − 2x + 5 for x = 1/2 (2 − √3)
Perform the indicated operations and express the result in simplified radical form with rationalized denominators. 2√3-7√14 3√3+ 2√14
One of the legs of a right triangle is 3√5 and the hypotenuse is 5√3. Find the area of the triangle.
The legs of a right triangle are √18 and √32. Find the perimeter of the triangle.
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. √3+ n(√3+ n = √n)-¹
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. (2√15 2 2 (2√15 2
Perform the indicated operations and express the result in simplified radical form with rationalized denominators.(√7 − 2√15)(3√7 − √15)
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. -2 a 1 b2
An idealized model of the thermodynamic process in a gasoline engine is the Otto cycle. The efficiency e of the process isShow that e T₁ T₂-T₁+T₂ r Tr r r T₂rr r
The monthly loan payment P for loan amount A with an annual interest rate r (as a decimal) for t years isFind the monthly payment for a $30,000 car loan if it is a 5-year loan with an annual interest rate of 2.6%. P = A(+2) 12 r 1 - ( 1 + 1/72) - 12t
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 2√3y-√2 9√3y - 5√2
Perform the indicated operations and express the result in simplest radical form with rationalized denominators.(1 + 61/2)(31/2 + 21/2)(31/2 − 21/2)
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. √46² +1
In fluid dynamics, the expressionarises. Combine terms in the denominator, rationalize the denominator, and simplify. a 1 + m/√r
Perform the indicated operations and express the result in simplest radical form with rationalized denominators.(x3n−1/xn−1)1/n
In optics, the combined focal length F of two lenses is given bywhere f1 and f2 are the focal lengths of the lenses and d is the distance between them. Simplify the right side of this equation. F = [fi¹ + f₂¹ + d(f₁f₂)-¹1¹,
Perform the indicated operations and express the result in simplest radical form with rationalized denominators.(3x1/2 − 2x)(3x1/2 + 2x)
Perform the indicated operations and express the result in simplest radical form with rationalized denominators.x3/8(x5/8 − 8x13/8)
The expressionoccurs in the study of electric circuits. Simplify this expression by combining terms under the radical and rationalizing the denominator. 2√R² 2L 4L C
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 2√x - a 3√x + 5a
The resonant frequency ω of a capacitance C in parallel with a resistance R and inductance L (see Fig. 11.9) isCombine terms under the radical, rationalize the denominator, and simplify. W JLC R²C L
In analyzing an electronic filter circuit, the expressionis used. Rationalize the denominator, expressing the answer without the fraction f0/f 8A π²√1 + (fo/f)²
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 7- 4√6 3 + 4√6
According to the Doppler effect, the frequency f0 observed of a sound of frequency f, when the observer is moving away from the source with velocity u, is given bywhere v is the velocity of sound. Simplify the right side. fo || fv V u SIE
When analyzing the ratio of resultant forces when forces F and T act on a structure, the expressionarises. Simplify this expression. √F² +T²/√F-² +T-²
Given that v = aptr, where v is the velocity of an object, a is its acceleration, and t is the time, use the metric units given in Exercise 74 to show that p = r = 1.Data from Exercises 74The metric units for the velocity v of an object are m · s−1, and the units for the acceleration a of the
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. √7 – 5 √5 √5 + 3√7
In analyzing a tuned amplifier circuit, the expressionis used. Rationalize the denominator. 20 2-1
When studying the orbits of Earth satellites, the expressionarises. Express it in simplest rationalized radical form. GM 47² 2 1/3 T2/3
The metric units for the velocity v of an object are m · s−1, and the units for the acceleration a of the object are m · s−2 . What are the units for v/a?
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 4 6-2√7
When studying a solar energy system, the units encountered are kg · s−1(m · s−2)2 . Simplify these units and include joules (see Exercise 71) and only positive exponents in the final result.Data from Exercises 71The metric unit of energy, the joule (J), can be expressed as kg · s−2 · m2 .
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. √2 √3 - 4√2
The electric current i (in A) in a circuit with a battery of voltage E, a resistance R, and an inductance L, iswhere t is the time after the circuit is closed. See Fig. 11.4. Find i for E = 6.20 V, R = 1.20Ω, L = 3.24 H, and t = 0.00100 s. (The number e is irrational and can be found from the
When analyzing the velocity of an object falling through a great distance, the expression a√2g/a arises. Show by rationalizing the denominator that this expression takes on a simpler form.
An approximate equation for the efficiency E (in percent) of an engine iswhere R is the compression ratio. Explain how this equation can be written with fractional exponents and then find E for R = 7.35. E = 100(11/5/R²
An expression used in determining the characteristics of a spur gear isRationalize the denominator. 50 50+ √V
The units for the electric quantity called permittivity are C2 · N−1 · m−2. Given that 1 F = 1 C2 · J−1, show that the units of permittivity are F / m.
For a heat-seeking rocket in pursuit of an aircraft, the distance d (in km) from the rocket to the aircraft is where θ is shown in Fig. 11.3. Find d for θ = 125.0°. d 500(sin 0)¹/2 (1 - cos 0) 3/2
In analyzing the tuning of an electronic circuit, the expression is used. Expand and simplify this expression. [w - ww-12
In designing musical instruments, the equationarises for the frequency of vibration of strings. Write this equation with a rationalized denominator. f = T V4μL²
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 10√a 4√a - 2c
The metric unit of energy, the joule (J), can be expressed as kg · s−2 · m2 . Simplify these units and include newtons and only positive exponents in the final result.
Perform the indicated operations and express the result in simplest radical form with rationalized denominators. 3x 2√3x - y
For an object oscillating at the end of a spring and on which there is a force that retards the motion, the equation m2 + bm + k2 = 0 must be solved. Here, b is a constant related to the retarding force, and k is the spring constant. By substitution, show thatis a solution. m = (√b² - 4k² - b)
The escape velocity required to overcome the gravitational pull of a planet or star isWrite this equation with a rationalized denominator. V esc 2GM VR
Among the products of a specialty furniture company are tables with tops in the shape of a regular octagon (eight sides). Express the area A of a table top as a function of the side s of the octagon
Living organisms contain a certain amount of carbon-14. After death, this amount decays according to the equation in Exercise 69 (this is used in carbon-14 dating). What percentage of an organism’s carbon-14 is still present 2580 years after death?Data from In Exercises 69Carbon-14 has a
Solve for x: 25x = 27 (22x)2 .
Carbon-14 has a half-life of approximately 5730 years. If A0 is the original amount present, then the amount present after t years is given by A(t) = A02−t/5730 , If 55.0 mg is present initially, how much is present after 3250 years?
Display the graphs of y1 = √x + 2 and y = √x + √2 on a calculator to show that √x + 2 is not equal to √x + √2.
Perform the indicated operations and express the result in simplest radical form with rationalized denominators.(√8a − 2√b)(√8a + 2√b)
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