Solution of Numerical Equations

Excerpts

Solution of Numerical Equations
Hence $-1$ is the remainder obtained when the given polynomial $x^3 - 2x - 5$ is divided by $x-2$.
Solution of Numerical Equations
Newton used the close approximation $0.1$ to $p$, in spite of the fact that this value exceeds the root $p$ and hence led to a negative correction at the next step.
Solution of Numerical Equations
Given an approximate value $a$ of a real root, we can usually find a closer approximation $a+h$ to the root by neglecting the powers $h^2$, $h^3, \dotsc$ of the small number $h$ in Taylor’s formula (§56)
Solution of Numerical Equations
W. G. Horner, London Philosophical Transactions, 1819. Earlier (1804) by P. Ruffini.
Solution of Numerical Equations
Failure is certain if we use a point $P_2$ such that a single bend point lies between it and $S$.
Solution of Numerical Equations
The advantage of having $c$ at each step is that we know a close limit of the error made in the approximation to the root.
Solution of Numerical Equations
To find the imaginary roots $x+yi$ of an equation $f(z)=0$ with real coefficients, expand $f(x+yi)$ by Taylor’s theorem;

Equations

Solution of Numerical Equations
p^3 + 6p^2 + 10p - 1 = 0
The transformed equation for p, obtained by putting x = 2 + p in x^3 - 2x - 5 = 0.
Solution of Numerical Equations
x^3 - 2x - 5 \equiv (x-2)^3 + 6(x-2)^2 + 10(x-2) - 1
An identity in x expressing the cubic x^3 - 2x - 5 in powers of x - 2, whose constant term -1 is the remainder on division by x - 2.
Solution of Numerical Equations
t^3 + 6.282t^2 + 11.154508t - 0.006153416 = 0
The transformed equation for the correction t to the root 2.094 of x^3 - 2x - 5 = 0, obtained by Horner's method.
Solution of Numerical Equations
f(a+h) = f(a) + f'(a)h + f''(a) \frac{h^2}{2} + \dotsb
Taylor's expansion of f(a+h) about a, with the higher powers of h indicated by dots.
Solution of Numerical Equations
f(a) + f'(a)h = 0
Neglecting the powers h^2, h^3, ... of the small correction h leaves this linear condition on h.
Solution of Numerical Equations
h = \frac{-f(a)}{f'(a)}
Newton's method takes the correction to the approximate root a as minus f(a) divided by f'(a); a + h is the next approximation.
Solution of Numerical Equations
f'(a) = \tan XTP
Geometrically the derivative at a point equals the tangent of the angle the tangent line makes with the x-axis, the tangent being drawn at P on the graph.
Solution of Numerical Equations
c = \frac{\alpha f(\beta) - \beta f(\alpha)}{f(\beta) - f(\alpha)}
The abscissa c where the chord joining the points at alpha and beta meets the x-axis, lying between alpha and beta.
Solution of Numerical Equations
-f(\alpha) : c - \alpha = f(\beta) : \beta - c
By similar triangles, the chord AB cuts the x-axis at c so that the two segments on the axis are in the same ratio as the two ordinates.
Solution of Numerical Equations
f'(x) = 4x^3 + 3x^2 - 6x - 1
The derivative of f(x) = x^4 + x^3 - 3x^2 - x - 4.
Solution of Numerical Equations
d^4 + 9d^3 + 27d^2 + 31d + 6 = 0
The transformed equation for d obtained by putting x = 2 + d in x^4 + x^3 - 3x^2 - x - 4 = 0.
Solution of Numerical Equations
6.3r^2 + 11.16196r + 0.000541708 = 0
Newton's transformed equation for the second correction r, after neglecting the cubic term in q.
Solution of Numerical Equations
q^3 + 6.3q^2 + 11.23q + 0.061 = 0
Newton's transformed equation for the correction q to the approximation 0.1 of p, from x = 2 + 0.1 + q.
Solution of Numerical Equations
x - \sin x = \tfrac{1}{4} \pi
The condition that the chord cuts off a segment of one-eighth the circle's area, reduced to an equation in the central angle x.
Solution of Numerical Equations
\tfrac{1}{2} r^2(x - \sin x) = \tfrac{1}{8} \pi r^2
The area of a circular segment with central angle x equals one-eighth of the area of the circle.
Solution of Numerical Equations
h = \frac{-f(a)}{f'(a)} = \frac{-a + \sin a + \tfrac{1}{4} \pi}{1 - \cos a}
Newton's correction for the root of x - sin x - pi/4 = 0, written with f(a) = a - sin a - pi/4 and f'(a) = 1 - cos a.
Solution of Numerical Equations
f'(x) = 2 - \frac{M}{x}
Derivative of f(x) = 2x - log x - 7 where log x = M log_e x.
Solution of Numerical Equations
\log x = M \log_e x
A common logarithm equals the natural logarithm multiplied by the modulus M, which is about 0.4343.
Solution of Numerical Equations
x^3 - 2x - 5 = 0
The equation whose real root between 2 and 3 is computed by Horner's method, and the reference point for the transformed equations.
Solution of Numerical Equations
f(x) - f''(x) \frac{y^2}{1·2} + f''''(x) \frac{y^4}{1·2·3·4} - \dotsb = 0
The real part of the Taylor expansion of f(x+yi) set to zero, giving the first of the two real equations for an imaginary root.
Solution of Numerical Equations
f'(x) - f'''(x) \frac{y^2}{1·2·3} + f^{(5)}(x)\frac{y^4}{5!} - \dotsb = 0
The imaginary part of the Taylor expansion of f(x+yi) set to zero, giving the second real equation for an imaginary root.
Solution of Numerical Equations
x^4 - x + 1 - 6x^2 y^2 + y^4 = 0
The real equation obtained from z^4 - z + 1 = 0 with z = x + yi, by setting the real part to zero.
Solution of Numerical Equations
4x^3 - 1 - 4xy^2 = 0
The imaginary-part equation obtained from z^4 - z + 1 = 0 with z = x + yi.
Solution of Numerical Equations
y^2 = x^2 - \frac{1}{4x}
Solving the imaginary-part equation for y^2 in terms of x.
Solution of Numerical Equations
-4x^6 + x^2 + \frac{1}{16} = 0
The cubic equation in x^2 obtained by eliminating y^2 between the two real equations for the imaginary roots of z^4 - z + 1.

Problems

Exercise Page89

Exercise Page89, problem 1, p. 89
$x^3 +2x+20=0$.
Printed answer:
- Single, $-2.46955$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -2.46955
Exercise Page89, problem 10, p. 89
The real cube root of $7.976$.
Printed answer:
- $1.997997997$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
7.976 ENTER 1 ENTER 3 ÷ yˣ
```
Calculator: +1997997996659985299027791689850350E-33; the book prints 1.997997997. Run on the calculator core at firmware 628c96c.
Exercise Page89, problem 11, p. 89
The abscissa of the real point of intersection of the conics $y=x^2$, $xy+x+3y-6=0$.
Printed answer:
- $1.094551482$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 1.094551482}
Exercise Page89, problem 12, p. 89
Find to 3 decimal places the abscissas of the points of intersection of $x^2+y^2=9$,$y=x^2-x$.
Printed answer:
- $2.059$, $-1.228$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread [2.059, -1.228]
Exercise Page89, problem 13, p. 89
A sphere two feet in diameter is formed of a kind of wood a cubic foot of which weighs two-thirds as much as a cubic foot of water (i.e., the specific gravity of the wood is $2/3$). Find to four significant figures the depth $h$ to which the floating sphere will sink in water.
Printed answer:
- $1.2261$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 1.2261
Exercise Page89, problem 14, p. 89
If the specific gravity of cork is $1/4$, find to four significant figures how far a cork sphere two feet in diameter will sink in water.
Printed answer:
- $0.6527 = \text{reciprocal of } 2 \cos 40°$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 0.6527
Exercise Page89, problem 15, p. 89
Compute $\cos 20°$ to four decimal places by use of 3A = 4^3 A - 3A, 60° = 12.
Printed answer:
- $0.9397$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise Page89, problem 16, p. 89
Three intersecting edges of a rectangular parallelopiped are of lengths $6$, $8$, and $10$ feet. If the volume is increased by $300$ cubic feet by equal elongations of the edges, find the elongation to three decimal places.
Printed answer:
- $1.3500$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 1.3500
Exercise Page89, problem 17, p. 89
Given that the volume of a right circular cylinder is $\alpha\pi$ and the total area of its surface is $2\beta\pi$, prove that the radius $r$ of its base is a root of $r^3 - \beta r + \alpha = 0$. If $\alpha = 56$, $\beta = 28$, find to four decimal places the two positive roots $r$. The corresponding altitude is $\alpha/r^2$.
Printed answer:
- $2.7138$, $3.3840$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2.7138, 3.3840]
Exercise Page89, problem 18, p. 89
What rate of interest is implied in an offer to sell a house for $2700 cash, or in annual installments each of $1000 payable 1, 2, and 3 years from date?
Printed answer:
- $5.46\%$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {r: 0.0546}
Exercise Page89, problem 19, p. 89
Find the rate of interest implied in an offer to sell a house for $3500 cash, or in annual installments each of $1000 payable 1, 2, 3, and 4 years from date.
Printed answer:
- $5.57\%$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem {r: 5.57}
Exercise Page89, problem 2, p. 89
$x^3 +3x^2 -2x-5=0$.
Printed answer:
- $-1.20164$, $1.33006$, $-3.12842$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-1.20164, 1.33006, -3.12842]
Exercise Page89, problem 20, p. 89
Find the rate of interest implied in an offer to sell a house for $3500 cash, or $4000 payable in annual installments each of $1000, the first payable now.
Printed answer:
- $9.70\%$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {r: 0.0970}
Exercise Page89, problem 3, p. 89
$x^3 +x^2 -2x-1=0$.
Printed answer:
- $1.24698$, $-1.80194$, $-0.44504$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1.24698, -1.80194, -0.44504]
Exercise Page89, problem 4, p. 89
$x^4 +4x^3 -17.5x^2 -18x+58.5=0$.
Printed answer:
- $± 2.1213203$, $\Neg2.1231056$, $-6.1231056$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2.1213203, -2.1213203, 2.1231056, -6.1231056]
Exercise Page89, problem 5, p. 89
$x^4 -11,727x+40,385=0$.
Printed answer:
- $3.45592$, $21.43067$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3.45592, 21.43067]
Exercise Page89, problem 6, p. 89
$x^3 =10$.
Printed answer:
- $2.15443$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2.15443
Exercise Page89, problem 7, p. 89
$x^3 +4x^2 -7=0$.
Printed answer:
- $-1.7728656$, $\Neg1.1642479$, $-3.3913823$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-1.7728656, 1.1642479, -3.3913823]
Exercise Page89, problem 8, p. 89
$x^3 -7x-7=0$.
Printed answer:
- $\Neg3.0489173$, $-1.3568958$, $-1.6920215$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3.0489173, -1.3568958, -1.6920215]
Exercise Page89, problem 9, p. 89
The root between $2$ and $3$ of $x^3 -x-9=0$ (make only 3 transformations).
Printed answer:
- $2.24004099$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2.24004099

Exercise Page94

Exercise Page94, problem 1, p. 94
For $f(x) = x^4 + x^3 - 3x^2 - x - 4$, show by Descartes’ rule of signs that $f'(x)=0$ and $f''(x)=0$ each have a single positive root and that neither has a root between $1$ and $2$. Which of the values $1$ and $2$ should be taken as $\beta$?
Printed answer:
- $2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page94, problem 2, p. 94
When seeking a root between $2$ and $3$ of $x^3 - x - 9 = 0$, which value should be taken as $\beta$?
Printed answer:
- $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page96

Exercise Page96, problem 1, p. 96
Find to 8 decimal places the root between $2$ and $3$ of $x^3 - x - 9 = 0$.
Printed answer:
- $2.24004099$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2.24004099
Exercise Page96, problem 2, p. 96
Find to 7 decimal places the root between $2$ and $3$ of $x^3 - 2x^2 - 2 = 0$.
Printed answer:
- $2.3593041$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2.3593041
Exercise Page96, problem 3, p. 96
Find the real cube root of $7.976$ to 6 decimal places.
Printed answer:
- $1.997998$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
7.976 ENTER 3 GOLD ˣ√y
```
Calculator: +1997997996659985299027791689850351E-33; the book prints 1.997998. Run on the calculator core at firmware 628c96c.
Exercise Page96, problem 4, p. 96
Explain by Taylor’s expansion of $f(2+d)$ why the values of f(2), f’(2), 12f”(2), 12·3 f”’(2), 12·3·4 f””(2) are in reverse order the coefficients of the transformed equation d^4 + 9d^3 + 27d^2 + 31d + 6 = 0, obtained in the Example in the text, and printed in heavy type.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page96, problem 5, p. 96
The method commonly used to find the positive square root of $n$ by a computing machine consists in dividing $n$ by an assumed approximate value $a$ of the square root and taking half the sum of $a$ and the quotient as a better approximation. Show that the latter agrees with the value of $a+h$ given by applying Newton’s method to $f(x) = x^2-n$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page98

Exercise Page98, problem 1, p. 98
$\frac{1}{4}$.
Printed answer:
- $132° 20.7'$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 132 + Rational(69, 200)
Exercise Page98, problem 10, p. 98
Find $x$ to 5 decimal places in $x = 3\log_e x$.
Printed answer:
- $1.85718$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1.85718
Exercise Page98, problem 2, p. 98
$\frac{3}{8}$.
Printed answer:
- $157° 12'$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 157 + Rational(1, 5)
Exercise Page98, problem 3, p. 98
Solve $2x - \log x = 9$.
Printed answer:
- $4.8425364$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 4.8425364
Exercise Page98, problem 4, p. 98
Solve $3x - \log x = 9$.
Printed answer:
- $3.1668771$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 3.1668771
Exercise Page98, problem 5, p. 98
Find the angle just $>15°$ for which $\frac{1}{2} \sin x + \sin 2x = 0.64$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- solve: no printed answer to check
Exercise Page98, problem 6, p. 98
Find the angle just $>72°$ for which $x - \frac{1}{2} \sin x = \frac{1}{4} \pi$.
Printed answer:
- $72° 17'$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem (72 + Rational(17, 60))*pi/180
Exercise Page98, problem 7, p. 98
Find all solutions of Ex. 5 by replacing $\sin 2x$ by $2\sin x\cos x$, squaring, and solving the quartic equation for $\cos x$.
Printed answer:
- $15° 16\tfrac{1}{2}'$, $85° 56\tfrac{1}{2}'$, $212° 49'$, $225° 57'$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem [611/40, 10313/120, 12769/60, 13557/60]
Exercise Page98, problem 8, p. 98
Solve similarly $\sin x + \sin 2x = 1.2$.
Printed answer:
- $5° 56\tfrac{1}{2}'$, $25° 18'$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem [5 + Rational(113, 120), 25 + Rational(3, 10)]
Exercise Page98, problem 9, p. 98
Find $x$ to 6 decimal places in $\sin x = x - 2$.
Printed answer:
- $2.5541949$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 2.5541949

Exercise Page99

Exercise Page99, problem 1, p. 99
$z^3 - 2z - 5 = 0$.
Printed answer:
- $-1.04727± 1.13594 i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["-1.04727 + 1.13594*I", "-1.04727 - 1.13594*I"]
Exercise Page99, problem 2, p. 99
$28z^3 + 9z^2 - 1 = 0$.
Printed answer:
- $-\frac{2}{7} ± \frac{1}{7}\sqrt{3}i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["-2/7 + sqrt(3)*I/7", "-2/7 - sqrt(3)*I/7"]
Exercise Page99, problem 3, p. 99
$z^4 - 3z^2 - 6z = 2$.
Printed answer:
- $-1±i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["-1 + I", "-1 - I"]
Exercise Page99, problem 4, p. 99
$z^4 - 4z^3 + 11z^2 - 14z + 10 = 0$.
Printed answer:
- $1±i$, $1±2i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["1 + I", "1 - I", "1 + 2*I", "1 - 2*I"]
Exercise Page99, problem 5, p. 99
$z^4 - 4z^3 + 9z^2 - 16z + 20 = 0$. Hint: E(x) x(x - 2)(16x^4 - 64x^3 + 136x^2 - 144x + 65) = 0, and the last factor becomes $(w^2 + 1)(w^2 + 9)$ for $2x = w + 2$.
Printed answer:
- $2±i$, $±2i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["2 + I", "2 - I", "2*I", "-2*I"]

Exercise Page100

Exercise Page100, problem 1, p. 100
What arc of a circle is double its chord?
Printed answer:
- $217° 12' 27.4'' = 3.790988$ radians.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 3.790988
Exercise Page100, problem 10i, p. 100
$4 \tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its roots when (i) $\tau = 0.002$ and (ii) $\tau = 0.99$.
Printed answer:
- (i) $0.327739$, $0.339224$, $1124.333037$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [0.327739, 0.339224, 1124.333037]
Exercise Page100, problem 10ii, p. 100
$4 \tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its roots when (i) $\tau = 0.002$ and (ii) $\tau = 0.99$.
Printed answer:
- (ii) $0.250279$, $0.894609$, $1.127839$. % [** PP: If above, LaTeX thinks next token is an optional argument] [Set $x = 1 + y$, $y = 1/z$ and solve by trigonometry.]
verified: the printed answer passed a computed check
How it was checked
- solve: passes [0.250279, 0.894609, 1.127839]
Exercise Page100, problem 11, p. 100
Solve $x^x = 100$.
Printed answer:
- $3.597285$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3.597285
Exercise Page100, problem 12, p. 100
Solve $x = 10\log x$.
Printed answer:
- $10$, $1.371288$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: FLAG-PARSE [10, 1.371288]
Exercise Page100, problem 13, p. 100
Solve $x + \log x = x \log x$.
Printed answer:
- $0.326878$, $12.267305$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: FLAG-PARSE [0.326878, 12.267305]
Exercise Page100, problem 14, p. 100
Solve Kepler’s equation $M = x - e \sin x$ when $M = 332° 28' 54.8''$, $e = 14° 3' 20''$.
Printed answer:
- $324° 16' 29.55''$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem (324 + 16/60 + 29.55/3600)*pi/180
Exercise Page100, problem 15, p. 100
In what time would a sum of money at 6% interest compounded annually amount to as much as the same sum at simple interest at 8%?
Printed answer:
- $10$ yr. $4$ mo. $0$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"t": 10.333406}
Exercise Page100, problem 16, p. 100
In a semicircle of diameter $x$ is inscribed a quadrilateral with sides $a$, $b$, $c$, $x$; then $x^3 - (a^2 + b^2 + c^2) x - 2abc = 0$ (I. Newton). Given $a = 2$, $b = 3$, $c = 4$, find $x$.
Printed answer:
- $6.074674$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 6.074674
Exercise Page100, problem 17, p. 100
What rate of interest is implied in an offer to sell a house for $9000 cash, or $1000 down and $3000 at the end of each year for three years?
Printed answer:
- $6.13$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 0.0613
Exercise Page100, problem 2, p. 100
What arc of a circle is double the distance from the center of the circle to the chord of the arc?
Printed answer:
- $42° 20' 47\tfrac{1}{4}''$ doubled.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 2*(42 + 20/60 + 47.25/3600)*pi/180
Exercise Page100, problem 3, p. 100
If $A$ and $B$ are the points of contact of two tangents to a circle of radius unity from a point $P$ without it, and if arc $AB$ is equal to $PA$, find the length of the arc.
Printed answer:
- $133° 33.8'$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem (133 + 33.8/60)*pi/180
Exercise Page100, problem 4, p. 100
Find the angle at the center of a circle of a sector which is bisected by its chord.
Printed answer:
- $108° 36' 14''$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem (108 + 36/60 + 14/3600)*pi/180
Exercise Page100, problem 5, p. 100
Find the radius of the smallest hollow iron sphere, with air exhausted, which will float in water if its shell is $1$ inch thick and the specific gravity of iron is $7.5$.
Printed answer:
- $21.468212$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"R": 21.468212}
Exercise Page100, problem 6, p. 100
From one end of a diameter of a circle draw a chord which bisects the semicircle.
Printed answer:
- Angle at center $47° 39' 13''$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page100, problem 7, p. 100
The equation $x \tan x = c$ occurs in the theory of vibrating strings. Its approximate solutions may be found from the graphs $y = \cot x$, $y = x/c$. Find $x$ when $c = 1$.
Printed answer:
- $49° 17' 36.5''$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem (49 + 17/60 + 36.5/3600)*pi/180
Exercise Page100, problem 8, p. 100
The equation $\tan x = x$ occurs in the study of the vibrations of air in a spherical cavity. From an approximate solution $x_1 = 1.5\pi$, we obtain successively better approximations $x_2 = \tan^{-1} x_1 = 1.4334 \pi$, $x_3 = \tan^{-1} x_2, \dotsc$. Find the first three solutions to 4 decimal places.
Printed answer:
- $1.4303\pi$, $2.4590\pi$, $3.4709\pi$; $257° 27' 12.225''$ more exact than first.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["1.4303*pi", "2.4590*pi", "3.4709*pi"]
Exercise Page100, problem 9, p. 100
Find to 3 decimal places the first five solutions of x = 2x2-x^2, which occurs in the theory of vibrations in a conical pipe.
Printed answer:
- $x/\pi = 0.6625, 1.891, 2.930, 3.948, 4.959$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["0.6625*pi", "1.891*pi", "2.930*pi", "3.948*pi", "4.959*pi"]