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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
A drill bit 3/8 in. in diameter rotates at 1200 r/min. What is the linear velocity of a point on its circumference?
Insert the proper sign, > or < or =, between the given expressions. Explain your answers.cos 360° 2cos 180°
Evaluate the given trigonometric functions directly, without first changing to degree measure. Round answers to three significant digits.sin (−1.34)
Determine the values of the given trigonometric functions directly on a calculator. The angles are approximate.cos (−162.32°)
A helicopter blade is 2.75 m long and is rotating at 420 r/min. What is the linear velocity of the tip of the blade?
Insert the proper sign, > or < or =, between the given expressions. Explain your answers.tan 180° tan 0°
Evaluate the given trigonometric functions directly, without first changing to degree measure. Round answers to three significant digits.cot (−4.86)
Most DVD players use constant linear velocity. This means that the angular velocity of the disk continually changes, but the linear velocity of the point being read by the laser remains the same. If a DVD rotates at 1590 r/min for a point 2.25 cm from the center, find the number of revolutions per
Insert the proper sign, > or < or =, between the given expressions. Explain your answers.sin 270° 3sin 90°
Solve the given quadratic equations by factoring.(x + a)2 − b2 = 0
Without solving the given equations, determine the character of the roots.3.6t2 + 2.1 = 7.7t
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.7 + 3C = −2C2
Find the equation of the parabola that contains the points (−1, 14)(1, 9), and (2, 8).
Solve the given quadratic equations by factoring.bx2 − b = x − b2x
Without solving the given equations, determine the character of the roots.0.45s2 + 0.33 = 0.12s
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.5y = 4y2 − 8
The vertical distance d (in cm) of the end of a robot arm above a conveyor belt in its 8-s cycle is given by d = 2t2 − 16t + 47. Sketch the graph of d = f (t).
Solve the given quadratic equations by factoring.x2 + 2ax = b2 − a2
Solve the given problems. All numbers are accurate to at least two significant digits.Find k if the equation x2 + 4x + k = 0 has a real double root.
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.a2x2 + 2ax + 2 = 0
When mineral deposits form a uniform coating 1 mm thick on the inside of a pipe of radius r (in mm), the cross-sectional area A through which water can flow is A = π(r2 − 2r + 1). Sketch A = f(r).
Solve the given quadratic equations by factoring.x2(a2 + 2ab + b2) = x(a + b)
Solve the given problems. All numbers are accurate to at least two significant digits.Find the smallest positive integer value of k if the equation x2 + 3x + k = 0 has roots with imaginary numbers.
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.16r2 = 8r − 1
The shape of the Gateway Arch in St. Louis can be approximated by the parabola y = 192 − 0.0208x2 (in meters) if the origin is at ground level, under the center of the Arch. Display the equation representing the Arch on a calculator. How high and wide is the Arch?
In Eq. (7.1), for a = 2,b = −7, and c = 3, show that the sum of the roots is −b/a.Eq.7.1ax2 + bx + c = 0
Solve the given problems. All numbers are accurate to at least two significant digits.Solve the equation x4 − 5x2 + 4 = 0 for x.
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.ay2 = a − 3y
Under specified conditions, the pressure loss L (in lb/in.2per 100 ft), in the flow of water through a fire hose in which the flow is q gal/min, is given by L = 0.0002q2 + 0.005q. Sketch the graph of L as a function of q, for q < 100 gal/min.
For the equation of Exercise 41 show that the product of the roots is c/a.Data from Exercises 41In Eq. (7.1), for a = 2,b = −7, and c = 3, show that the sum of the roots is −b/a.Eq.7.1ax2 + bx + c = 0
Solve the given problems. All numbers are accurate to at least two significant digits.Without drawing the graph or completely solving the equation, explain how to find the number of x-intercepts of a quadratic function.
Solve the given quadratic equations by any appropriate algebraic method. If there are no real roots, state this as the answer.2bx = x2 − 3b
A computer analysis of the power P (in W) used by a pressing machine shows that P = 50i − 3i2, where i is the current (in A). Sketch the graph of P = f(i).
In finding the dimensions of a crate, the equation 12x2 − 64x + 64 = 0 is used. Solve for x, if x > 2.
Solve the given problems. All numbers are accurate to at least two significant digits.Use the discriminant b2 − 4ac to determine if the equation 90x 2 − 123x + 40 = 0 can be solved by factoring. Explain why or why not. Do not solve.
Solve the given quadratic equations by completing the square.x2 − x − 30 = 0
Solve the given problems. All numbers are accurate to at least two significant digits.Solve 6x2 − x = 15 for x by (a) Factoring, (b) Completing the square,(c) The quadratic formula. Which is (i) Longest?(ii) Shortest?
Tests show that the power P (in hp) of an automobile engine as a function of r (in r/min) is given by P = −5.0 × 10−6r2 + 0.050r − 45(1500 < r < 6000 r/min). Sketch the graph of P vs. r and find the maximum power that is produced.
If a rocket is launched with an initial velocity of 320 ft/s, its height above ground after t seconds is given by −16t2 + 320t (in ft). Find the times when the height is 0.
Solve the given equations. x - 4 x - 1 2 X
Solve the given quadratic equations by completing the square.x2 = 2x + 5
Solve the given problems. All numbers are accurate to at least two significant digits.In machine design, in finding the outside diameter D0 of a hollow shaft, the equation D02 − DD0 − 0.25D2 = 0 is used. Solve for D0 if D = 3.625 cm.
The height h (in m) of a fireworks shell shot vertically upward as a function of time t (in s) is h = −4.9t2 + 68t + 2. How long should the fuse last so that the shell explodes at the top of its trajectory?
The voltage V across a semiconductor in a computer is given by V = αI + βI2 , where I is the current (in A). If a 6-V battery is conducted across the semiconductor, find the current if α = 2Ω and β = 0.5 Ω/A.
Solve the given equations. V-1 3 SE +1
Solve the given quadratic equations by completing the square.2t2 = t + 4
Solve the given problems. All numbers are accurate to at least two significant digits.A missile is fired vertically into the air. The distance s (in ft) above the ground as a function of time t (in s) is given by s = 300 + 500t − 16t2. (a) When will the missile hit the ground? (b) When
In a certain electric circuit, the resistance R (inΩ) that gives resonance is found by solving the equation 25R = 3(R2 + 4). Solve this equation graphically (to 0.1Ω).
The mass m (in Mg) of the fuel supply in the first-stage booster of a rocket is m = 135 − 6t − t2 , where t is the time (in s) after launch. When does the booster run out of fuel?
Solve the given quadratic equations by completing the square.4x2 − 8x = 3
Solve the given problems. All numbers are accurate to at least two significant digits.In analyzing the deflection of a certain beam, the equation 8x2 − 15Lx + 6L2 = 0 is used. Solve for x, if x < L.
If the radius of a circular solar cell is increased by 1.00 cm, its area is 96.0 cm2. What was the original radius?
Solve the given equations. x² - 3x x - 3 x2² x + 2
The power P (in MW) produced between midnight and noon by a nuclear power plant is P = 4h2 − 48h + 74, where h is the hour of the day. At what time is the power 664 MW?
A security fence is to be built around a rectangular parking area of 20, 000 ft2. If the front side of the fence costs $20/ft and the other three sides cost $10/ft, solve graphically for the dimensions (to 1 ft) of the parking area if the fence is to cost $7500. See Fig. 7.19. $20/ft $10/ft 20,000
Solve the given problems. All numbers are accurate to at least two significant digits.A homeowner wants to build a rectangular patio with an area of 20.0 m2, such that the length is 2.0 m more than the width. What should the dimensions be?
The diagonal of a rectangular floor is 3.00 ft less than twice the length of one of the sides. If the other side is 15.0 ft long, what is the area of the floor?
Solve the given equations. x-2 x-5 15 x2 – 5x
In determining the speed s (in mi/h) of a car while studying its fuel economy, the equation s2 − 16s = 3072 is used. Find s.
Solve the given problems. All numbers are accurate to at least two significant digits.Two cars leave an intersection at the same time, one going due east and the other due south. After one has gone 2.0 km farther than the other, they are 6.0 km apart on a direct line. How far did each go?
Find a quadratic equation for which the solutions are 0.5 and 2.
Solve the given problems. All numbers are accurate to at least two significant digits.A student cycled 3.0 km/h faster to college than when returning, which took 15 min longer. If the college is 4.0 km from home, what were the speeds to and from college?
In calculating the current in an electric circuit with an inductance L, a resistance R, and a capacitance C, it is necessary to solve the equation Lm2 + Rm + 1/C = 0. Solve for m in the terms of L, R, and C. See Fig. 7.7. Fig.7.7 L elle L ти R НЕ C
An airplane pilot could decrease the time t (in h) needed to travel the 630 mi from Ottawa to Milwaukee by 20 min if the plane’s speed v is increased by 40 mi/h. Set up the appropriate equation and solve graphically for v (to two significant digits).
Find a quadratic equation for which the solutions are a and b.
Sketch the graphs of the given functions by using the vertex, the y-intercept, and one or two other points.y = 2x2 − x − 1
In finding the radius r of a circular arch of height h and span b, we use the formula shown below. Solve for h. r = b² + 4h² 8h
Solve the given problems. All numbers are accurate to at least two significant digits.For a rectangle, if the ratio of the length to the width equals the ratio of the length plus the width to the length, the ratio is called the golden ratio. Find the value of the golden ratio, which the ancient
Solve the given equations involving fractions. 2- 1 3 x x + 2
Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation.x3 − x = 0
Solve the given equations involving fractions. 1 x - 3 2
Sketch the graphs of the given functions by using the vertex, the y-intercept, and one or two other points.y = −4x2 − 1
When focusing a camera, the distance r the lens must move from the infinity setting is given by r = f 2/(p − f ) , where p is the distance from the object to the lens, and f is the focal length of the lens. Solve for f.
Solve the given equations involving fractions. 1 2x 3 4 1 2x + 3
Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation.x3 − 4x2 − x + 4 = 0
Sketch the graphs of the given functions by using the vertex, the y-intercept, and one or two other points.y = x − 3x2
Sketch the graphs of the given functions by using the vertex, the y-intercept, and one or two other points.y = 2x2 + 8x − 10
Solve the given equations involving fractions. 2 1 x - 3 = 3 =
Solve the given equations by using a calculator. Round solutions to the nearest hundredth. If there are no real roots, state this as the answer.2x2 + x − 18 = 0
A computer monitor has a viewing screen that is 33.8 cm wide and 27.3 cm high, with a uniform edge around it. If the edge covers 20.0% of the monitor front, what is the width of the edge?
The spring constant k is the force F divided by the amount x the spring stretches (k = F/x). See Fig. 7.2(a). For two springs in series [see Fig. 7.2(b)], the reciprocal of the spring constant kc for the combination equals the sum of the reciprocals of the individual spring constants. Find the
Solve the given problems. All numbers are accurate to at least two significant digits.In order to have the proper strength, the angle iron shown in Fig. 7.8 must have a cross-sectional area of 53.5 cm2. Find the required thickness x. Fig. 7.8 15.6 cm X 10.4 cm X
Solve the given equations by using a calculator. Round solutions to the nearest hundredth. If there are no real roots, state this as the answer.−4s2 − 5s = 1
An investment of $2000 is deposited at a certain annual interest rate. One year later, $3000 is deposited in another account at the same rate. At the end of the second year, the accounts have a total value of $5105.45. The interest rate r can be found by solving 2000(1 + r)2 + 3000(1 + r) =
The combined resistance R of two resistances R1 and R2 connected in parallel [see Fig. 7.3(a)] is equal to the product of the individual resistances divided by their sum. If the two resistances are connected in series [see Fig. 7.3(b)], their combined resistance is the sum of their individual
Solve the given equations by using a calculator. Round solutions to the nearest hundredth. If there are no real roots, state this as the answer.3x2 = −x − 2
In remodeling a house, an architect finds that by adding the same amount to each dimension of a 12-ft by 16-ft rectangular room, the area would be increased by 80 ft2. How much must be added to each dimension?
Solve the given equations by using a calculator. Round solutions to the nearest hundredth. If there are no real roots, state this as the answer.x(15x − 12) = 8
Solve the given problems. All numbers are accurate to at least two significant digits.Two pipes together drain a wastewater-holding tank in 6.00 h. If used alone to empty the tank, one takes 2.00 h longer than the other. How long does each take to empty the tank if used alone?
A quadratic equation f (x) = 0 has a solution x = 2. Its graph has its vertex at (−1, 6). What is the other solution?
Solve the given problems. All numbers are accurate to at least two significant digits.The length of a tennis court is 12.8 m more than its width. If the area of the tennis court is 262 m2, what are its dimensions? See Fig. 7.9. Fig. 7.9 w + 12.8 m A = 262 m² W
A hydrofoil made the round-trip of 120 km between two islands in 3.5 h of travel time. If the average speed going was 10 km/h less than the average speed returning, find these speeds.
Find c such that y = 2x2 + 16x + c has exactly one real root.
Solve the given problems. All numbers are accurate to at least two significant digits.For an optical lens, the sum of the reciprocals of p, the distance of the object from the lens, and q, the distance of the image from the lens, equals the reciprocal of f, the focal length of the lens. If p is 5.0
Set up the appropriate quadratic equations and solve.A rectangular solar panel is 20 cm by 30 cm. By adding the same amount to each dimension, the area is doubled. How much is added?
Solve the given quadratic equations by any appropriate method. All numbers are accurate to at least two significant digits.The bending moment M of a simply supported beam of length L with a uniform load of w kg/m at a distance x from one end is M = 0.5wLx − 0.5wx2. For what values of x is M = 0?
Two circular oil spills are tangent to each other. If the distance between centers is 800 m and they cover a combined area of 1.02 × 106m2 , what is the radius of each? See Fig. 7.10. Fig. 7.10 - 1.02 x 106 m² 800 m
Solve the given quadratic equations by any appropriate method. All numbers are accurate to at least two significant digits.A nuclear power plant supplies a fixed power level at a constant voltage. The current I (in A) is found by solving the equation I2 − 17I − 12 = 0. Solve for I > 0.
Replace the A with the proper expression such that the fractions are equivalent. 2R R+T 2R²T A
Perform indicated operations and simplify. 1-x3+x бу 4у
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