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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

In studying the frictional effects on a flywheel, the revolutions per minute R that it makes as a function of the time t (in min) is given by R = 4520(0.750)2.50t. Find t for R = 1950 r/min.
The power gain of an electronic device such as an amplifier is defined as n = 10 log(P0/Pi), where n is measured in decibels, P0 (in W) is the power output, and Pi (in W) is the power input. If P0 =
The magnitudes (visual brightnesses), m1 and m2 , of two stars are related to their (actual) brightnesses, b1 and b2, by the equation m1 − m2 = 2.5 log(b2/b1). As a result of this definition,
An equation used to calculate the capacity C (in bits/s) of a telephone channel is C = Blog2 (1 + R). Solve for R.
Taking into account the weight loss of fuel, the maximum velocity vm of a rocket is vm = u(ln m0 − ln ms) − gtf, where m0 is the initial mass of the rocket and fuel, ms is the mass of the rocket
An original amount of 100 mg of radium radioactively decomposes such that N mg remain after t years. The function relating t and N is t = 2350(ln 100 − ln N). Sketch the graph.
A computer analysis of the luminous efficiency E (in lumens/W) of a tungsten lamp as a function of its input power P (in W) is given by E = 22.0(1 − 0.65e−0.008P). Sketch the graph of E as a
An approximate formula to predict the population (in millions) of the United States t years after January 1, 2020 is P = 329e0.00711t. According to this formula, in what year will the U.S. population
A state lottery pays $500 for a $1 ticket if a person picks the correct three-digit number determined by the random draw of three numbered balls. To find the number of drawings x required to have a
The amount A of alcohol in a person’s bloodstream is given by A = A0ekt, where A0 is an initial amount (mg alcohol/mL blood), t is the time (in h) after drinking the alcohol, and k is a constant
The amount A of cesium-137 (a dangerous radioactive element) remaining after t years is given by A = A0 (0.5t/30.3),where A0 is the initial amount. In what year will the cesium-137 be 10% of
The current i (in μA) in a microchip circuit is i = 25(2−2500t), where t is the time (in ns) after the current is turned off. How long does it take the current to be 0.25 μA?
If A0 dollars are invested in a mutual fund that has an annual yield of r (in %), the value of A after n years is A = A0(1 + r)n .(a) Solve for n. (b) How many years does it take A to double if
The temperature T (in °C) of the coffee in a cup t min after being heated in a microwave was T = 22.5 + 76.0e−0.15t . Display the graph of this function on a calculator.
The bending moment M (in N · m) of a particular concrete column is given by ln M = 15.34. What is the value of M?
Use logarithms to find the x-intercept of the graph of y = 2 − 52x−3.
Use a calculator to solve the equation log5 x = 2x − 7.
If x logb 7 = and y logb 2, = express logb 42 in terms of x and y.
Evaluate: 7(10log 0.1) + 6000(100log 0.001).
Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line y = x on a graphing calculator and note that each is the mirror image of the
For a radioactive isotope, the constant of proportionality k is given by Cobalt-60, used in cancer therapy, has a halflife of 5.27 years. What is the value of k? k = log, (2) half-life
Evaluate: log(2 log1000).
If f(x) = 2 logb x and f(8) = 3, find f(4).
When a tractor-trailer turns a right angle corner, its rear wheels follow a curve called a tractrix, the equation for which isDisplay the curve on a graphing calculator. y = log 1 +
Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line y = x on a graphing calculator and note that each is the mirror image of the
Solve for x: 2x + 32(2−x) = 12.
Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line y = x on a graphing calculator and note that each is the mirror image of the
Evaluate 3√ln e8 − √log104
Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line y = x on a graphing calculator and note that each is the mirror image of the
Use a calculator to verify that 3 ln 2 + 0.5 ln 64 = 3 ln 4.
If log N = 1.513 and ln P = 3.796, find N + P.
In Exercise 47, the graph of y = log0.5 x is plotted. By inspecting the graph and noting the properties of log 0.5 x, describe some of the differences of logarithms to a base less than 1 from those
Evaluate the given expressions using the property blogbx = x.5(102 log 3)
Evaluate the given expressions using the property blogbx = x.3e2 ln2
The time t (in ps) required for N calculations by a certain computer design is t = N + log2N. Sketch the graph of this function.
Evaluate the given expressions using the property blogbx = x.2eln 7.5
An equation used in measuring the flow of water in a channel is C = −a log10 (b/R). Solve for R.
Evaluate the given expressions using the property blogbx = x.10log4
The work W done by a sample of nitrogen gas during an isothermal (constant temperature) change from volume V1 to volume V2 is given by W = k loge (V2/V1). Solve for V1.
Plot the graphs of the given functions on log-log paper.xy4 = 16
The capacitance C of a cylindrical capacitor is given by C = k/loge (R2/R1) , where R1 and R2 are the inner and outer radii. Solve for R1.
In analyzing the power gain in a microprocessor circuit, the equation N = 10(2 log10I1 − 2 log10I2 + log10R1 − log10R2) is used. Express this with a single logarithm on the right side.
The time t (in years) for an investment to double iswhere r is the annual yield rate. How long does it take an investment to double if r = (a) 3.00%, (b) 6.00%, (c) 9.00%? In 2 In(1 +
Plot the graphs of the given functions on log-log paper.y = 3√x
An equation relating the number N of atoms of radium at any time t in terms of the number N0 of atoms at t = 0 is loge (N/N0) = −kt, where k is a constant. Solve for N.
A container of water is heated to 90°C and then placed in a room at 0°C. The temperature T of the water is related to the time t (in min) by loge T = loge 90.0 − 0.23t. Find T as a function of t.
Plot the graphs of the given functions on semilogarithmic paper.y = 5x3
The velocity v of a rocket when its fuel is completely burned is given by v = u loge (w0/w), where u is the exhaust velocity, w0 is the liftoff weight, and w is the burnout weight. Solve for w.
On the same screen of a calculator, display the graphs ofWhat conclusion can be drawn from the display? Y₁ = log10x log10 (x² + 1) and y2 X logio 2+1
When a person ingests a medication capsule, it is found that the rate R (in mg/min) that it enters the bloodstream in time t (in min) is given by log10 R − log10 5 = t log10 0.95. Solve for R as a
Plot the graphs of the given functions on semilogarithmic paper.y = 8x
The magnitudes (visual brightness), m1 and m2, of two stars are related to their (actual) brightness's, b1 and b2, by the equation m1 − m2 = 2.5 log10 (b2/b1). Solve for b2.
A study of urban density shows that the population density D (in persons/mi2) is related to the distance r (in mi) from the city center by loge D = loge a − br + cr2, where a, b, and c are positive
Solve the given equation.log(n + 2) + log n = 0.4771
The curve in which a uniform wire or rope hangs under its own weight is called a catenary. An example of a catenary that we see every day is a wire strung between utility poles, as shown in Fig.
The use of the insecticide DDT was banned in the United States in 1972. A computer analysis shows that an expression relating the amount A still present in an area, the original amount A , 0 and the
A computer analysis of sales of a product showed that the net profit p (in $1000) during a given year was p = 2 ln(t + 2) − 0.5t, where t is in months. Graphically determine the date when p = 0.
Solve the given equation.2 log3 2 − log3(x + 1) = log3 5
Find the domain of the function f(x) = loge (2 − x).
The peak current I m (in A) in an alternating-current circuit is given bywhere P is the power developed, Z is the magnitude of the impedance, and ∅ is the phase angle between the current and
Solve the given equation.log8(x + 2) = 2 − log8 2
Solve the given systems of equations algebraically.2x² + y² = 22y² = x² + 8
In the study of spur gears in contact, the equationis used. Solve for r12 kC = √R² R² + √√r² = r² - A
Solve the given equation. √n + 4 + 2√√n + 2 = 3
The distance d (in km) to the horizon from a height h (in km) above the surface of Earth isFind h for d = 980 km. d = √1.28 x 104h + h².
A formula used in calculating the range R for radio communication isSolve for h. R = √√2rh+ h².
A workbench is in the shape of a trapezoid, as shown in Fig. 14.14. If the perimeter of the workbench is 260 cm, what is its area if y > x? Fig. 14.14 x X y 65.0 cm
Solve the given equation. √x + 1 + √√√x = 2
Solve the given equation. √3x² 2 √√√x² + 7 = 1
A jet travels at 610 mi/h relative to the air. It takes the jet 1.6 h longer to travel the 3660 mi from London to Washington, D.C., against the wind than it takes from Washington to London with the
Solve the given equation. 2√5x3-1 = 2x
A rectangular play area is twice as long as it is wide. If the area is 648 m2, what are its dimensions?
The resonant frequency f in an electric circuit with an inductance L and a capacitance C is given bySolve for L. f = 1 2T LC
Solvealgebraically. Then compare the solution with that of Example 4. Noting that the algebraic steps after isolating the radical are identical, why is the solution different?Data from Example 4Solve
Two guy wires, one 140 ft long and the other 120 ft long, are attached at the same point of a TV tower with the longer one secured in the (level) ground 30 ft farther from the base of the tower than
Solve the given equation. 33√x - 2 = 9
Solve the given equation. √5x + 9 + 1 = x
A roof truss in the shape of a right triangle has a perimeter of 90 ft. If the hypotenuse is 1 ft longer than one of the other sides, what are the sides of the truss?
Iffind x if f(x + 6) = 5. f(x) = √x + 3,
An open liner for a carton is to be made from a rectangular sheet of cardboard of area 216 in.2 by cutting equal 2.00-in. squares from each corner and bending up the sides. If the volume within the
A rectangular TV screen has an area of 1540 in.2 and a diagonal of 60.0 in. Find the dimensions of the screen.
Solve the given equation. 3√2Z + 4 = 2Z
Security fencing encloses a rectangular storage area of 1600 m2 that is divided into two sections by additional fencing parallel to the shorter sides. Find the dimensions of the storage area if
A special washer is made from a circular disc 3.50 cm in radius by removing a rectangular area of 12.0 cm2 from the center. If each corner of the rectangular area is 0.50 cm from the outer edge of
Solve the given equations algebraically and check the solutions with a graphing calculator. √2x + 1 = √√√x + 4 = 1
A set of equal electrical resistors in series has a total resistance (the sum of the resistances) of 78.0 Ω. Another set of two fewer equal resistors in series also has a total resistance of 78.0
In the theory dealing with optical interferometers, the equation √F = 2√p/(1 − p) is used. Solve for p if F = 16.
In a certain roller mechanism, the radius of one steel ball is 2.00 cm greater than the radius of a second steel ball. If the difference in their masses is 7100 g, find the radii of the balls. The
Solve the given equation.(x2 + 5x)2 − 5(x2 + 5x) = 6
An equation used in the study of the dispersion of light is μ = A + Bλ−2 + Cλ−4 . Solve for λ.
Two people, one walking 1.0 km/h faster than the other, are on straight roads that are perpendicular. After meeting at an intersection, each continues on straight. How fast is each walking if they
Solve the given equations algebraically and check the solutions with a graphing calculator. √2x + 1 + 3√√x = 9
The equivalent resistance RT of two resistors R1 and R2 in parallel is given by RT-1 = R1-1 + R2-1 . T If RT = 1.00 Ω and R2 =R1, find R1 and R2.
Solve the given equation. 4 r²+1 + 7 2r² +1. 2
A roof truss is in the shape of a right triangle. If there are 4.60 m of lumber in the truss and the longest side is 2.20 m long, what are the lengths of the other two sides of the truss?
Solve the given equations algebraically and check the solutions with a graphing calculator. √x - 2 + 3 = x
A paper drinking cup in the shape of a cone is constructed from 6π in.2 of paper. If the height of the cone is 4 in., find the radius.
Solve the given equation. √10 + 3√x
The impedance Z in an alternating-current circuit is 2.00Ω. If the resistance R is numerically equal to the square of the reactance X, find R and X. Use Z2 = R2 + X2
A circular hot tub is located on the square deck of a home. The side of the deck is 24 ft more than the radius of the hot tub, and there are 780 ft2 of deck around the tub. Find the radius of the hot

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