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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

It is known that of all plane curves that enclose a given area, the circle has the least perimeter. Show that if a plane curve of perimeter L encloses an area A then 4πA ≤ L2. Verify this inequality for a square and a semicircle.
Regulations for taking a party of school children on a trip abroad requires that the ratio of accompanied teachers to children travelling must be 1:6. If 54 children sign up for the trip, how many teachers must travel with the party?
The arithmetic–geometric inequalityBy applying the arithmetic–geometric inequality to the first two terms of this inequality, deduce that implies x + y 2 x+y 2 > xy >xy Use the substitution x = (a + b), y = 2(c + d), where a, b, c and d > 0, to show that and hence that ( + b )( + d ) = (a + b
Express the sets (a) {x: |x – 3|< 5, x in R} (b) {x: |x + 2| ≤ 3, x in R} as intervals.
Show that (√5 + √13)2 > 34 and determine without using a calculator the larger of √5 + √13 and √3 + √19.
The ratio of males to females in a class is 7:5. If there are 60 children in the class, how many are females?
Prove that for any two positive numbers x and y, the arithmetic–geometric inequality 1/2(x + y) ≥ √(xy) holds.
Show that if a < b, b > 0 and c > 0 thenObtain a similar inequality for the case a > b. a b a+c b + c
Show the following sets on number lines and express them as intervals: (a) {x|x - 4 | 6} (c) {x:|2x - 1| < 7} (b) (x:x + 3
A pipe has the form of a hollow cylinder as shown in Figure 1.4. Find its mass when(a) Its length is 1.5 m, its external diameter is 205 mm, its internal diameter is 160mm and its density is 5500 kgm–3;(b) Its length is l m, its external diameter is Dmm, its internal diameter is d mm and its
A prize of £7500 is to be divided among John, Jane and Mary in the ratios 3:4:8. How much will each receive?
(a) If n = n1 + n2 + n3 show that(This represents the number of ways in which n objects may be divided into three groups containing respectively n1, n2 and n3 objects.)(b) Expand the following expressions (2)(". n n n + Mz M n! n! n! nz!
Show the following intervals on number lines and express them as sets in the form {x: |ax + b| < c} or {x: |ax + b|≤ c}:(a) (1, 7) (b) [–4, –2](c) (17, 26)(d) [–1/2, 3/4]
Prove that ab = 1/4 [(a + b)2 – (a – b)2] Given 702 = 4900 and 362 = 1296, calculate 53 × 17.
(a) Evaluate(b) A square grid of dots may be divided up into a set of L-shaped groups as illustrated in Figure 1.35.How many dots are inside the third L shape?How many extra dots are needed to extend the 3 by 3 square to one of side 4 by 4? How many dots are needed to extend an (r – 1) by (r
Given that a In each case either prove that the statement is true or give a numerical example to show it can be false.If, additionally, a, b, c and d are all greater than zero, how does that modify your answer? (a) a-c
Show that (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Find the equations of the straight line(a) Which passes through the points (–6, –11) and (2, 5);(b) Which passes through the point (4, –1) and has gradient 1/3;(c) Which has the same intercept on the y axis as the line in (b) and is parallel to the line in (a).
Verify that (x + p)2 + q – p2 = x2 + 2px + q and deduce that ax + bx + c = ax + + + c = a[x 2 b 2a + C. b 4a
The average speed for a journey is the distance covered divided by the time taken.(a) A journey is completed by travelling for the first half of the time at speed v1 and the second half at speed v2. Find the average speed va for the journey in terms of v1 and v2.(b) A journey is completed by
Simplify the following expressions: (a) x x x 4 (d) x/3 Xx5/3 (i) -4 (b) x = x + (e) (4x)-1/2 (2) Va(x ) (2) (5x-20) (5x1/3 2 1. 2x1/2 x 1/2 x-/2 (k) (4ab)-3/2 (c) (x)-4 3 (f) (2x (j) (ab)/2 (ab-)
Find the equation of the circle which touches the y axis at the point (0, 3) and passes through the point (1, 0).
Express as a single fraction (a) (b) 12 2 3 3 1 2 (x + 1)(x + 2) x+1 + 3 x+2
Find the centres and radii of the following circles: (a) x + y + 2x - 4y + 1 = 0 (b) 4x - 4x + 4y + 12y + 9 = 0 (c) 9x + 6x +9y - 6y = 25
Factorize(a) x2y – xy2(b) x2yz – xy2z + 2xyz2(c) ax – 2by – 2ay + bx(d) x2 + 3x – 10(e) x2 – 1/4y2 (f) 81x4 – y4
Use the method of completing the square to manipulate the following quadratic expressions into the form of a number + (or –) the square of a term involving x.(a) x2 + 3x – 7 (b) 5 – 4x – x2(c) 3x2 – 5x + 4 (d) 1 + 2x – 2x2
For each of the two parabolasdetermine(a) The coordinates of the vertex,(b) The coordinates of the focus,(c) The equation of the directrix,(d) The equation of the axis of symmetry. Sketch each parabola. (i) y = 8x + 4y - 12, and (ii) x + 12y + 4x = 8
Factorize xz + 2yz – 2y – x.
Find the coordinates of the centre and foci of the ellipse with equationWhat are the coordinates of its vertices and the equations of its directrices? Sketch the ellipse. 25x + 16y 16y 100x256y + 724 = 0 -
An isosceles trapezium has non-parallel sides of length 20 cm and the shorter parallel side is 30 cm, as illustrated in Figure 1.8. The perpendicular distance between the parallel sides is h cm. Show that the area of the trapezium is h(30 + √(400 – h2))cm2. Figure 1.8 30 cm h cm 20 cm
Factorize the expressions(a) x2 +12x + 35(b) 2x2 + 9x – 5
An open container is made from a sheet of cardboard of size 200mm × 300mm using a simple fold, as shown in Figure 1.9. Show that the capacity Cml of the box is given by C = x(150 – x)(100 – x)/250Figure 1.9 200 mm 300 mm x mm x mm
Find the duodecimal equivalent of the decimal number 10.386 23.
Expand(a) (2x + 3y)2 (b) (2x – 3)3(c) 2x - X 4
Rearrange the following quadratic expressions by completing the square.(a) x2 + x – 12 (b) 3 – 2x + x2(c) (x – 1)2 – (2x – 3)2 (d) 1 + 4x – x2
Show that if y = x1/2 then the relative error bound of y is one-half that of x. Hence complete the table in Figure 1.36. a Va b b C Vc d Vd Correctly rounded values Value Figure 1.36 7.01 2.6476 52.13 0.01011 5.631 X 10 Absolute error bound Va Vb Vc Vd 2.65 0.005 0.0009 Relative error bound 0.0007
Rearrange the following formula to make s the subject m = P d s+t S-t
A hollow cone of base diameter 100mm and height 150mm is held upside down and completely filled with a liquid. The liquid is then transferred to a hollow circular cylinder of base diameter 80 mm. To what height is the cylinder filled?
Givenfind t in terms of u and x. 1-zx 1+zx = n
Assuming that all the numbers given are correctly rounded, calculate the positive root together with its error bound of the quadratic equation 1.4x2 + 5.7x – 2.3 = 0 Give your answer also as a correctly rounded number.
A dealer bought a number of equally priced articles for a total cost of £120. He sold all but one of them, making a profit of £1.50 on each article with a total revenue of £135. How many articles did he buy?
The quantities f, u and v are connected byFind f when u = 3.00 and v = 4.00 are correctly rounded numbers. Compare the error bounds obtained for f when(a) It is evaluated by taking the reciprocal of the sum of the reciprocals of u and v,(b) It is evaluated using the formula 1 f 11 1 1 u + V
Solve for t 1 1-t 1 1+t = 1
Using the method of completing the square (1.5a), obtain the formula for finding the roots of the general quadratic equation ax2 + bx + c = 0 (a ≠ 0)Data from 1.5(a) (a + b) = a + 2ab + b (1.5a)
Iffind the positive value of c when x = 4, y = 6, V1 = 120, V2 = 315. 3c + 3xc + x 3c + 3yc + y || yV xV
Find the values of x for which 1 3-x
A milk carton has capacity 2 pints (1136 ml).It is made from a rectangular waxed card using the net shown in Figure 1.37. Show that the total area A (mm2) of card used is given by A(h, w) = (2w + 145)(h + 80)with hw = 113 600/7. Show thatwith equality when 160w = 145h. Hence show that the minimum
Solve for p the equation 2p+1 p-1 p+5 p+1 + = 2
Find the values of x such that x2 + 2x + 2 > 50.
A family of straight lines in the x – y plane is such that each line joins the point (–p, p) on the line y = –x to the point (10 – p, 10 – p) on the line y = x, as shown in Figure 1.38, for different values of p. On a piece of graph paper, draw the lines corresponding to p = 1, 2, 3, ·
A food manufacturer found that the sales figure for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20 000 and that the number sold decreased by 100 for every 1p increase in its price.
A rectangle has a perimeter of 30 m. If its length is twice its breadth, find the length.
Find numbers A, B and C such that x x-1 = Ax + B + C x-1' x = 1
Find the numbers A, B and C such that x2 + 2x – 35 = A(x – 1)2 + B(x – 1) + C.
Find the values of x for which(a) 5/2 (b) 1/2 - x (c)(d) 3x - 2 x-1 >2
(a) A4 paper is such that a half sheet has the same shape as the whole sheet. Find the ratio of the lengths of the sides of the paper.(b) Foolscap paper is such that cutting off a square whose sides equal the shorter side of the paper leaves a rectangle which has the same shape
Given a0 = 1, a1 = 5, a2 = 2, a3 = 7, a4 = –1 and b0 = 0, b1 = 2, b2 = –2, b3 = 11, b4 = 3, calculate 03 (2) k=0 3 ) j=2 3 ygo 3 (0) I=y A (d) b k=0
Find the values of x for which x2 < 2 + |x|.
Evaluate(a) 4! (b) 3! × 2! (c) 6! (d) 7!/(2! × 5!)
Find the values of A and B such that (a) 1 (x+1)(x - 2) = (b) 3x + 2= A(x - (c) 5x+1 (x + x + 1) A B + x + 1 x-2 1) + B(x - 2) A(2x + 1) + B (x + x + 1)
Prove that(a) x2 + 3x – 10 ≥ – (7/2)2(b) 18 + 4x – x2 ≤ 22(c) x + 4/x ≥ 4 where x > 0
In how many ways can the letters of the word REGAL be arranged in a line, and in how many of those do the two letters A and E appear in adjacent positions?
Expand the expression (2 + x)5.
Find the values of A, B and C such that 2x2 – 5x + 12 = A(x – 1)2 + B(x – 1) + C
Given a0 = 2, a1 = –1, a2 = –4, a3 = 5, a4 = 3 and b0 = 1, b1 = 1, b2 = 2, b3 = –1, b4 = 2, calculate (2) ' k=0 2 (c) Sabi k=1 3 (b) = 1 4 (d) j=0
Find the equation of the straight line that passes through the points (1, 2) and (3, 3).
Evaluate(a) 5! (b) 3!/4! (c) 7!/(3! × 4!)(d)(e)(f) 5 2
Find the equation of the straight line passing through the point (3, 2) and parallel to the line 2y = 3x + 4. Determine its x and y intercepts.
Find the equation of the circle with centre (1, 2) and radius 3.
Find the equation of the straight line(a) With gradient 3/2 passing through the point (2, 1);(b) With gradient –2 passing through the point (–2, 3);(c) Passing through the points (1, 2) and (3, 7);(d) Passing through the points (5, 0) and (0, 3);(e) Parallel to the line 3y – x = 5, passing
Write down the equation of the circle with centre (1, 2) and radius 5.
Find the equation of the circle which passes through the points (0, 0), (0, 2), (4, 0).
Find the radius and the coordinates of the centre of the circle with equation x2 + y2 + 4x – 6y = 3
Find the point of intersection of the line y = x – 1 with the circle x2 + y2 – 4y – 1 = 0.
Find the equation of the circle with centre (–2, 3) that passes through (1, –1).
Find the equation of the tangent at the point (2, 1) of the circle x2 + y2 – 4y – 1 = 0.
Find the equation of the circle that passes through the points (1, 0), (3, 4) and (5, 0).
A point P moves in such a way that its total distance from two fixed points A and B is constant. Show that it describes an ellipse.
Find the equation of the tangent to the circle x2 + y2 – 4x – 1 = 0 at the point (1, 2).
A point moves in such a way that its distance from a fixed point F is equal to its perpendicular distance from a fixed line. Show that it describes a parabola.
A rod, 50 cm long, moves in a plane with its ends on two perpendicular wires. Find the equation of the curve followed by its midpoint.
(a) Find the equation of the tangent at the point (1, 1) to the parabola y = x2. Show that it is parallel to the line through the points (1/2, 1/4), (3/2, 9/4), which also lie on the parabola.(b) Find the equation of the tangent at the point (a, a2) to the parabola y = x2. Show that it is parallel
The feet of the altitudes of triangle A(0, 0), B(b, 0) and C(c, d) are D, E and F respectively. Show that the altitudes meet at the point O(c, c(b – c)/d). Further, show that the circle through D, E and F also passes through the midpoint of each side as well as the midpoints of the lines AO, BO
Express the number 150.4152(a) Correct to 1, 2 and 3 decimal places; (b) Correct to 1, 2 and 3 significant figures.
Find the coordinates of the focus and the equation of the directrix of the parabola whose equation is 3y2 = 8x The chord which passes through the focus parallel to the directrix is called the latus rectum of the parabola. Show that the latus rectum of the above parabola has length 8/3.
Compute(a) 3.142 + 4.126 (b) 5.164 – 2.341 (c) 235.12 × 0.531Calculate estimates for the effects of rounding errors in each answer and give the answer as a correctly rounded number.
For the ellipse 25x2 + 16y2 = 400 find the coordinates of the foci, the eccentricity, the equations of the directrices and the lengths of the semi-major and semi-minor axes.
Give the absolute and relative error bounds of the following correctly rounded numbers(a) 29.92 (b) –0.01523 (c) 3.9 × 1010
For the hyperbola 9x2 – 16y2 = 144 find the coordinates of the foci and the vertices and the equations of its asymptotes.
Evaluate 13.92 × 5.31 and 13.92 ÷ 5.31. Assuming that these values are correctly rounded numbers, calculate error bounds for each answer and write them as correctly rounded numbers which have the greatest possible number of significant digits.
EvaluateAssuming that all the values given are correctly rounded numbers, calculate an error bound for your answer and write it as a correctly rounded number. 6.721- 4.931 x 71.28 89.45
State the numbers of decimal places and significant figures of the following correctly rounded numbers:(a) 980.665 (b) 9.11 × 10–28(c) 2.9978 × 1010 (d) 2.00 × 1033(e) 1.759 × 107 (f) 6.67 × 10–8
A continuous belt of length L m passes over two wheels of radii r and R m with their centres a distance l m apart, as illustrated in Figure 1.32. The belt is sufficiently tight for any sag to be negligible. Show that L is given approximately byFind the error inherent in this approximation and
In a right-angled triangle the height is measured as 1 m and the base as 2 m, both measurements being accurate to the nearest centimetre. Using Pythagoras’ theorem, the hypotenuse is calculated as 2.236 07 m. Is this a sensible deduction? What other source of error will occur?
A cable company is to run an optical cable from a relay station, A, on the shore to an installation, B, on an island, as shown in Figure 1.33. The island is 6 km from the shore at its nearest point, P, and A is 9 km from P measured along the shore. It is proposed to run the cable from A along the
Determine the error bound and relative error bound for x, where(a) x = 35 min ± 5 s(b) x = 35 min ± 4%(c) x = 0.58 and x is correctly rounded to 2dp.
A value is calculated to be 12.9576, with a relative error bound of 0.0003. Calculate its absolute error bound and give the value as a correctly rounded number with as many significant digits as possible.
Using exact arithmetic, compute the values of the expressions below. Assuming that all the numbers given are correctly rounded, find absolute and relative error bounds for each term in the expressions and for your answers. Give the answers as correctly rounded numbers.(a) 1.316 – 5.713 + 8.010(b)
Evaluatefor a = 4.99 and b = 5.01. Give absolute and relative error bounds for each answer. a+b, ab, ax b, alb
Complete the table below for the computationand give the result as the correctly rounded answer with the greatest number of significant figures. 9.21 (3.251 3.115)/0.112 +

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