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mathematics
basic technical mathematics

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

The velocity ν of an Earth satellite varies directly as the square root of its mass m, and inversely as the square root of its distance r from the center of Earth. If the mass is halved and the distance is doubled, how is the speed affected?
Hooke’s law states that the force needed to stretch a spring is proportional to the amount the spring is stretched. If 10.0 lb stretches a certain spring 4.00 in, how much will the spring be stretched by force of 6.00 lb? See Fig. 18.8. Fig. 18.8 4.00 in. 10.00 lb 6.00 lb
Given that 104 cm2 = 106 mm2, what area in square centimeters is 2.50 ×105 mm2?
A total of 322 bolts are in two containers. The ratio of the number of bolts in the first container to the number in the second container is 5/9. How many are in each container?
The flow of water Q through a fire hose is proportional to the cross-sectional area A of the hose. If 250 gal flows through a hose of diameter 2.00 in. in a given time, how much would flow through a hose 3.00 in. in diameter in the same time?
How many meters per second are equivalent to 45.0 km/h?
Give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values.y varies directly as the square of x, and y = 27 when x = 3.
How many gallons per hour are equivalent to 540 mL/min?
The rate H of heat removal by an air conditioner is proportional to the electric power input P. The constant of proportionality is the performance coefficient. Find the performance coefficient of an air conditioner for which H = 1.8 kW and P = 720 W.
Give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values.f varies inversely as l, and f = 500 when l = 800.
A particular type of automobile engine produces 62,500 cm3 of carbon monoxide in 2.00 min. How much carbon monoxide is produced in 45.0 s?
The energy E available daily from a solar collector varies directly as the percent p that the sun shines during the day. If a collector provides 1200 kJ for 75% sunshine, how much does it provide for a day during which there is 35% sunshine?
Give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values.v is directly proportional to x and inversely proportional to the cube of y, and v = 10 when x = 5 and y = 4.
For the lever balanced at the fulcrum, the relationholds, where F1 and F2 are forces on opposite sides of the fulcrum at distances L1 and L2 (see Fig. 18.11). Find L2 if F1 = 4.50 lb, F2 = 6.75 lb, and L1 = 17.5 in. F₁ F₂ L₂ L₁
An airplane consumes 36.0 gal of gasoline in flying 425 mi. Under similar conditions, how far can it fly on 52.5 gal?
The distance d that can be seen from horizon to horizon from an airplane varies directly as the square root of the altitude h of the airplane. If d = 133 mi for h = 12,000 ft, find d for h = 16,000 ft.
In a physics experiment, a given force was applied to three objects. The mass m and the resulting acceleration a were recorded as follows:(a) Is the relationship a = f(m) one of direct or inverse variation? Explain. (b) Find a = f(m). m (g) a (cm/s²) 2.0 3.0 4.0 30 20 15
Give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values.r varies jointly as u, v, and the square of w, and r = 1/8 when u = 1/2, v = 1/4, and w = 1/3.
After a race, a runner’s GPS watch shows the length of the race course to be 5058 m. If the actual length of the course is 4998 m, find the percent error in the watch’s reading. See Exercise 23.Data from Exercises 23The percent error in a measurement is the ratio of the error in the measurement
The image height h and object height H for the lens shown in Fig. 18.12 are related to the image distance q and object distance p byFind q ifh = 24 cm,H = 84 cm, andp = 36 cm. h 오 H P
The time t required to empty a wastewater-holding tank is inversely proportional to the cross-sectional area A of the drainage pipe. If it takes 2.0 h to empty a tank with a drainage pipe for which A = 48 in.2, how long will it take to empty the tank if A = 68 in.2?
The lift L of each of three model airplane wings of width w was measured and recorded as follows:If L varies directly as the square of w, find L = f(w). Does it matter which pair of values is used to find the constant of proportionality? Explain. w (cm) L (N) 20 40 40 60 10 40 90
The weight of a person on Earth and the weight of the same person on Mars are proportional. If an astronaut weighs 920 N on Earth and 350 N on Mars, what is the weight of another astronaut on Mars if the astronaut weighs 640 N on Earth?
The time t required to make a particular trip is inversely proportional to the average speed v. If a jet takes 2.75 h at an average speed of 520 km/h, how long will it take at an average speed of 620 km/h? Explain the meaning of the constant of proportionality.
A company finds that the volume V of sales of a certain item and the price P of the item are related byFind V2 if P1 = $8.00, P2 = $6.00, 1 = and V1 = 3000 per week.
Two separate sections of a roof have the same slope. If the rise and run on one section are, respectively, 3.0 m and 6.3 m, what is the run on the other section if its rise is 4.2 m?
For two pulleys connected by a belt, the relationholds, where d is the diameter of the pulley and n is the number of revolutions per unit time it makes. Find n2 if d1 = 4.60 in., d2 = 8.30 in., and n 18.0 r/min. d₁ d2 n₂ n₁
When a bullet is fired from a loosely held rifle, the ratio of the mass of the bullet to that of the rifle equals the negative of the reciprocal of the ratio of the velocity of the bullet to that of the rifle. If a 3.0 kg rifle fires a 5.0-g bullet and the velocity of the bullet is 300 m/s, what is
A TV screen has an area of 1160 in.2. A PIP (picture in picture) has an area that is one-third as large as the remaining area seen. What are the areas of the PIP and the remaining area on the screen?
The power P required to propel a ship varies directly as the cube of the speed s of the ship. If 5200 hp will propel a ship at 12.0 mi/h, what power is required to propel it at 15.0 mi/h?
If c/d is in inverse ratio to a/b, then a/b = d/c (see Exercises 29 and 30). The current i (in A) in an electric circuit is in inverse ratio to the resistance R (inΩ). If i = 0.25 mA when R = 2.8Ω, what is i when R = 7.2 Ω?Data from Exercises 29In an electric instrument called a “Wheatstone
An apartment owner charges rent R proportional to the floor area A of the apartment. Find the equation relating R and A if an apartment of 900 ft2 rents for $850/month.
The f-number lens setting of a camera varies directly as the square root of the time t that the film is exposed. If the f-number is 8 (written as f/8) for t = 0.0200 s, find the f-number for t = 0.0098 s.
The number r of aluminum cans that can be made by recycling n used cans is proportional to n. How many cans can be made from 50,000 used cans if r = 115 cans for n = 125 cans?
By weight, the ratio of chlorine to sodium in table salt is 35.46 to 23.00. How much sodium is contained in 50.00 kg of salt?
The force F needed to tighten a bolt varies as the length L of the wrench handle. See Fig. 18.13. If F = 250 N for L = 22 cm, and F = 550 N for L = 10 cm, determine the equation relating F and L. Fig. 18.13 L
The force F on the blade of a wind generator varies jointly as the blade area A and the square of the wind velocity v. Find the equation relating F, A, and v if F = 19.2 lb when A = 3.72 ft2 and v = 31.4 ft/s.
Under certain conditions, the rate of increase v of bacteria is proportional to the number N of bacteria present. Find v for N = 7500 bacteria, if v = 800 bacteria/h for N = 4000 bacteria.
How much 100% fire-fighting foam concentrate must be added to 1000 L of a 2.0% solution to make a 4.0% solution?
The escape velocity v a spacecraft needs to leave the gravitational field of a planet varies directly as the square root of the product of the planet’s radius R and its acceleration due to gravity g. For Mars and Earth, RM = 0.533Re and gM = 0.400ge. Find vM for Mars if ve = 11.2 km/s.
The index of refraction n of a medium varies inversely as the velocity of light v within it. For quartz, n = 1.46 and v = 2.05 × 108 m/s. What is n for a diamond, in which v = 1.24 × 108 m/s?
In testing for quality control, it was found that 17 of every 500 computer chips produced by a company in a day were defective. If a total of 595 defective parts were found, what was the total number of chips produced during that day?
The force F between two parallel wires carrying electric currents is inversely proportional to the distance d between the wires. If a force of 0.750 N exists between wires that are 1.25 cm apart, what is the force between them if they are separated by 1.75 cm?
The value V of a precious stone varies directly as the square of its weight w. What is the loss in value if a stone for which V = $27,000 is cut into two pieces with weights in the ratio of 2 to 1?
The TV revenue of $2.50 billion is divided in the ratio of 1.85 to 1.65 between the players and owners of a football league as a result of the contract between them. What does each receive?
The velocity v of a pulse traveling in a string varies directly as the square root of the tension T in the string. If the velocity of a pulse in a string is 450 ft/s when the tension is 20.0 lb, find the velocity when the tension is 30.0 lb.
The period T of a pendulum varies directly as the square root of its length L. If T = π/2 s for L = 2.00 ft, find T for L = 4.00 ft.
The ratio of the width to height of an HDTV screen is 16 to 9, and for an older traditional screen the ratio is 4 to 3. For a 60.0 in. (diagonal) screen of each type, which has the greater area, and by what percent?
The average speed s of oxygen molecules in the air is directly proportional to the square root of the absolute temperature T. If the speed of the molecules is 460 m/s at 273 K, what is the speed at 300 K?
The monthly payment p on a mortgage is directly proportional to amount A that is borrowed. If p = $635 for A = $100,000.00, find p if A = $345,000.
Of Earth’s water area, the Pacific Ocean covers 46.0%, and the Atlantic Ocean covers 23.9%. Together they cover a total of 2.53 ×108 km2. What is the area of each?
The time t required to test a computer memory unit varies directly as the square of the number n of memory cells in the unit. If a unit with 4800 memory cells can be tested in 15.0 s, how long does it take to test a unit with 8400 memory cells?
The velocity v (in ft/s) of water from a fire hose is directly proportional to the square root of the pressure p (in lb/in.2). If v = 120 ft/s for p = 100 lb/in.2, find v for p = 64 lb/in.2.
The weight of an object is the force F on the object due to gravity, where F = mg. State how g (the acceleration due to gravity) varies with respect to the mass mb of the spherical body on which the object lies, and the radius of that body (assume all of the mass is at its center). (See Example
The electric resistance R of a wire varies directly as its length l and inversely as its cross-sectional area A. Find the relation between resistance, length, and area for a wire that has a resistance of 0.200 Ω for a length of 225 ft and cross-sectional area of 0.0500 in.2.
The component of velocity vx of an object moving in a circle with constant angular velocity ω varies jointly with ω and sin ωt. If ω = π/6 rad/s, and vx = −4 π ft/s when t = 1.00 s, find vx when t = 9.00 s.
The general gas law states that the pressure P of an ideal gas varies directly as the thermodynamic temperature T and inversely as the volume V. If P = 610 kPa for V = 10.0 cm3 and T = 290 K, find V for P = 400 kPa and T = 400 K.
The charge C on a capacitor varies directly as the voltage V across it. If the charge is 6.3 μC with a voltage of 220 V across a capacitor, what is the charge on it with a voltage of 150 V across it?
Noting the general gas law in Exercise 54, if V remains constant,(a) express P as a function of T. (b) A car tire is filled with air at 10.0°C with a pressure of 225 kPa, and after a while the air temperature in the tire is 60.0°C. Assuming no change in volume, what is the pressure?Data from
The amount of natural gas burned is proportional to the amount of oxygen consumed. If 24.0 lb of oxygen is consumed in burning 15.0 lb of natural gas, how much air, which is 23.2% oxygen by weight, is consumed to burn 50.0 lb of natural gas?
The power P of a gas engine is proportional to the area A of the piston. If an engine with a piston area of 8.00 in.2 can develop 30.0 hp, what power is developed by an engine with a piston area of 6.0 in.2?
The velocity v of a jet of fluid flowing from an opening in the side of a container is proportional to the square root of the depth d of the opening. If the velocity of the jet from an opening at a depth of 1.22 m is 4.88 m/s, what is the velocity of a jet from an opening at a depth of 7.62 m? See
The power P in an electric circuit varies jointly as the resistance R and the square of the current I. If the power is 10.0 W when the current is 0.500 A and the resistance is 40.0Ω, find the power if the current is 2.00 A and the resistance is 20.0Ω.
The decrease in temperature at a point above a region is directly proportional to the altitude of the point above the region. If the temperature T on the ground in Toronto is 22°C and a plane 3.50 km above notes that the temperature is 1.0°C, what is the temperature at the top of the CN Tower,
The difference m1 − m2 in magnitudes (visual brightnesses) of two stars varies directly as the base 10 logarithm of the ratio b1/b2 of their actual brightnesses. For two particular stars, if b2 = 100b1 for m1 = 7 and m2 = 2, find the equation relating m1, m2, b1, and b2.
The distance d an object falls under the influence of gravity varies directly as the square of the time t of fall. If an object falls 64.0 ft in 2.00 s, how far will it fall in 3.00 s?
The power gain G by a parabolic microwave dish varies directly as the square of the diameter d of the opening and inversely as the square of the wavelength λ of the wave carrier. Find the equation relating G, d, and λ if G = 5.5 ×104 for d = 2.9 m and λ = 3.0 cm.
The tangent of the proper banking angle θ of the road for a car making a turn is directly proportional to the square of the car’s velocity v and inversely proportional to the radius r of the turn. If 7.75° is the proper banking angle for a car traveling at 20.0 m/s around a turn of radius 300
The kinetic energy E of a moving object varies jointly as the mass m of the object and the square of its velocity v. If a 5.00-kg object, traveling at 10.0 m/s, has a kinetic energy of 250 J, find the kinetic energy of an 8.00-kg object moving at 50.0 m/s.
The intensity I of sound varies directly as the power P of the source and inversely as the square of the distance r from the source. Two sound sources are separated by a distance d, and one has twice the power output of the other. Where should an observer be located on a line between them such that
In a particular computer design, N numbers can be sorted in a time proportional to the square of log N. How many times longer does it take to sort 8000 numbers than to sort 2000 numbers?
The x-component of the acceleration of an object moving around a circle with constant angular velocity ω varies jointly as cos ωt and the square of ω. If the x-component of the acceleration is −11.4 ft/s2 when t = 1.00 s for ω = 0.524 rad/s, find the x-component of the acceleration when t =
In an electric circuit containing an inductance L and a capacitance C, the resonant frequency f is inversely proportional to the square root of the capacitance. If the resonant frequency in a circuit is 25.0 Hz and the capacitance is 95.0 μF, what is the resonant frequency of this circuit if the
The acoustical intensity I of a sound wave is proportional to the square of the pressure amplitude P and inversely proportional to the velocity v of the wave. If I = 0.474 W/m2 for P = 20.0 Pa and v = 346 m/s, find I if P = 15.0 Pa and v = 320 m/s.
The rate of emission R of radiant energy from the surface of a body is proportional to the fourth power of the thermodynamic temperature T. Given that a 25.0-W (the rate of emission) lamp has an operating temperature of 2500 K, what is the operating temperature of a similar 40.0-W lamp?
A metal circular ring has a circular cross section of radius r. If R is the radius of the ring (measured to the middle of the cross section), the volume V of metal in the ring varies directly as R and the square of r. If V = 2550 mm3 for r = 2.32 mm and R = 24.0 mm, find V for r = 3.50 mm and R =
To cook a certain vegetable mix in a microwave oven, the instructions are to cook 4.0 oz for 2.5 min or 8.0 oz for 3.5 min. Assuming the cooking time t is proportional to some power (not necessarily an integer) of the weight w, use logarithms to find t as a function of w.
The frequency f of a radio wave is inversely proportional to its wavelength λ. The constant of proportionality is the velocity of the wave, which equals the speed of light. Find this velocity if a certain radio wave has a frequency of 90.9 MHz and a wavelength of 3.29 m.
The acceleration of gravity g on a satellite in orbit around Earth varies inversely as the square of its distance r from the center of Earth. If g = 8.7 m/s2 for a satellite at an altitude of 400 km above the surface of Earth, find g if it is 1000 km above the surface. The radius of the Earth is
If the weight w of an airplane varies directly as the cube of its length L, and w = 3520 lb for L = 42.0 ft, what would be the weight of a plane that is 50.0 ft long?
The thrust T of a propeller varies jointly as the square of the number n of revolutions per second and the fourth power of its diameter d. What is the effect on T if n is doubled and d is halved?
Using holography (a method of producing an image without using a lens), an image of concentric circles is formed. The radius r of each circle varies directly as the square root of the wavelength λ of the light used. If r = 3.56 cm for λ = 575 nm, find r if λ = 483 nm.
The range R of a stream of water from a fire hose varies jointly as the square of the initial velocity v0 and the sine of twice the angle θ from the horizontal from which it is released. See Fig. 18.16. A stream for which v0 = 35 m/s and θ = 22° has a range of 85 m. Find the range if
Kepler’s third law of planetary motion states that the square of the period of any planet is proportional to the cube of the mean radius (about the sun) of that planet, with the constant of proportionality being the same for all planets. Using the fact that the period of Earth is 1 year and its
The stopping distance d of a car varies directly as the square of the velocity v of the car when the brakes are applied. A car moving at 32 mi/h can stop in 52 ft. What is the stopping distance for the car if it is moving at 55 mi/h?
The load L that a helical spring can support varies directly as the cube of its wire diameter d and inversely as its coil diameter D. A spring for which d = 0.120 in. and D = 0.953 in. can support 45.0 lb. What is the coil diameter of a similar spring that supports78.5 lb and for which d = 0.156
The volume rate of flow R of blood through an artery varies directly as the fourth power of the radius r of the artery and inversely as the distance d along the artery. If an operation is successful in effectively increasing the radius of an artery by 25% and decreasing its length by 2%, by how
The safe, uniformly distributed load L on a horizontal beam, supported at both ends, varies jointly as the width w and the square of the depth d and inversely as the distance D between supports. Given that one beam has double the dimensions of another, how many times heavier is the safe load it can
It took a roofing crew 8.25 hours to install architectural shingles on a 415 ft2 section of roof. If the number of work hours is proportional to the area of the roof, how long will it take the crew to install the remaining 755 ft2 of roof?
The distance s that an object falls due to gravity varies jointly as the acceleration g due to gravity and the square of the time t of fall. The acceleration due to gravity on the moon is 0.172 of that on Earth. If a rock falls for t0 seconds on Earth, how many times farther would the rock fall on
The heat loss L through fiberglass insulation varies directly as the time t and inversely as the thickness d of the fiberglass. If the loss through 8.0 in. of fiberglass is 1200 Btu in 30 min, what is the loss through 6.0 in. in 1 h 30 min?
A quantity important in analyzing the rotation of an object is its moment of inertia I. For a ball bearing, the moment of inertia varies directly as its mass m and the square of its radius r. Find the general expression for I if I = 39.9 g · cm2 for m = 63.8 g and r = 1.25 cm.
In the study of polarized light, the intensity I is proportional to the square of the cosine of the angle θ of transmission. If I = 0.025 W/m2 for θ = 12.0°, find I for θ = 20.0°.
The force F that acts on a pendulum bob is proportional to the mass m of the bob and the sine of the angle θ the pendulum makes with the vertical. If F = 0.120 N for m = 0.350 kg and θ = 2.00°, find F for m = 0.750 kg and θ = 3.50°.
A fruit-packing company plans to reduce the size of its fruit juice can (a right circular cylinder) by 10% and keep the price of each can the same (effectively raising the price). The radius and the height of the new can are to be equally proportional to those of the old can. Write one or two
Change each number to polar form and then perform the indicated operations. Express the final result in rectangular and polar forms. Check by performing the same operation in rectangular form using a calculator.(√3 − j)−8
Explain why the product of a complex number and its conjugate is real and nonnegative.
Simplify each of the given expressions.(a) −j21 (b) (−j)21
Determine the rectangular form and the polar form of the complex number for which the graphical representation is shown in the given figure. 3.75 51.6° Imag. 0 Real
Solve for x: (1 + jx)2 = 1 + j − x2

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