Category Archives: Engineering Mathematics Blog

Maximum Load of a Member | Stress | Mechanics of Materials | 3rd Edition | Timothy Philpot | Problem P1.1

A stainless steel tube with an outside diameter of 60 mm and a wall thickness of 5 mm is used as a compression member. If the axial normal stress in the member must be limited to 200 MPa, determine the maximum load P that the member can support.

SOLUTION:

The cross-sectional area of the stainless steel tube is

$\displaystyle A=\frac{\pi }{4}\left(D^2-d^2\right)$

$\displaystyle A=\frac{\pi }{4}\left[\left(60\:mm\right)^2-\left(50\:mm\right)^2\right]$

$\displaystyle A=863.938\:mm^2$

The normal stress in the tube can be expressed as

$\displaystyle \sigma =\frac{P}{A}$

The maximum normal stress in the tube must be limited to 200 MPa. Using 200 MPa as the allowable normal stress, rearrange this expression to solve for the maximum load P.

$\displaystyle P_{max}=\sigma _{max}A$

$\displaystyle P_{max}=\left(200\:MPa\right)\left(863.938\:mm^2\right)$

$\displaystyle P_{max}=\left(200\:\frac{N}{mm^2}\right)\left(863.938\:mm^2\right)$

$\displaystyle P_{max}=172\:788\:N$

$\displaystyle P_{max}=172.8\:kN$

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 3

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

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SOLUTION:

\begin{align*} \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right) & =\lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x\right)+\lim\limits_{x\to \frac{\pi }{4}}\left(\sin\:x\right)\\ & =\tan\:\frac{\pi }{4}+\sin\:\frac{\pi }{4}\\ & =1+\frac{\sqrt{2}}{2}\\ & =\frac{2+\sqrt{2}}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 2

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

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SOLUTION:

\begin{align*} \lim_{x\to 3}\left(\frac{4x+2}{x+4}\right)& =\frac{\lim\limits_{x\to 3}\left(4x+2\right)}{\lim\limits_{x\to 3}\left(x+4\right)}\\ & =\frac{\lim\limits_{x\to 3}\left(4x\right)+\lim\limits_{x\to 3}\left(2\right)}{\lim\limits_{x\to 3}\left(x\right)+\lim\limits_{x\to 3}\left(4\right)}\\ & =\frac{4\cdot \lim\limits_{x\to 3}\left(x\right)+2}{3+4}\\ & =\frac{4\cdot 3+2}{3+4}\\ & =\frac{12+2}{7}\\ & =\frac{14}{7}\\ & =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 1

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PROBLEM:

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

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SOLUTION:

\begin{align*} \lim_{x\to 2}\left(x^2-4x+3\right)& = \lim_{x\to 2}\left(x^2\right)-\lim_{x\to 2}\left(4x\right)+\lim_{x\to 2}\left(3\right)\\ & =\left[\lim_{x\to 2}\left(x\right)\right]^2-4\lim_{x\to 2}\left(x\right)+3\\ & =\left(2\right)^2-4\left(2\right)+3\\ & =-1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 10

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PROBLEM:

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .

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SOLUTION:

Part A

\begin{align*} f\left[g\left(x\right)\right] & =\frac{4}{\left(x^2-3\right)+3}\\ & =\frac{4}{x^2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Part B

\begin{align*} g\left[f\left(x\right)\right] & =\left(\frac{4}{x+3}\right)^2-3\\ & =\frac{16}{\left(x+3\right)^2}-3\\ & =\frac{16-3\left(x+3\right)^2}{\left(x+3\right)^2}\\ & =\frac{16-3\left(x^2+6x+9\right)}{\left(x+3\right)^2}\\ & =\frac{16-3x^2-18x-27}{\left(x+3\right)^2}\\ & =\frac{-3x^2-18x-11}{\left(x+3\right)^2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 9

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PROBLEM:

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

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SOLUTION:

\begin{align*} \frac{f\left(h+3\right)-f\left(3\right)}{h} & =\frac{\left[3\left(h+3\right)^2-4\left(h+3\right)+1\right]-\left[3\left(3\right)^2-4\left(3\right)+1\right]}{h} \\ & =\frac{3\left(h^2+6h+9\right)-4h-12+1-16}{h}\\ & =\frac{3h^2+18h+27-4h-12+1-16}{h}\\ & =\frac{3h^2+14h}{h}\\ & =\frac{h\left(3h+14\right)}{h}\\ & =3h+14 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8

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PROBLEM:

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

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SOLUTION:

\begin{align*} \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h} & =\frac{\left[\left(x+h\right)^2+1\right]-\left(x^2+1\right)\:}{h}\\ \\ & =\frac{x^2+2xh+h^2+1-x^2-1}{h}\\ \\ & =\frac{2xh+h^2}{h}\\ \\ & =\frac{h\left(2x+h\right)}{h}\\ \\ & =2x+h \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 7

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PROBLEM:

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

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SOLUTION:

Refer to the figure below for an elevation view.

Schematic Diagram of a right circular cylinder inscribed in a right circular cone. — Diagram of a right circular cylinder with a base radius of r and height y inscribed in a right circular cone with base radius r and height h.

By ratio and proportion of two similar triangles, we have

\begin{align*} \frac{y}{r-x} & = \frac{h}{r} \\ y & =\frac{h\left(r-x\right)\:}{r} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 6

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PROBLEM:

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

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SOLUTION:

Let $S$ =stiffness, $b$ =breadth, and $d$ =depth

\begin{align*} S & =bd^3 \\ S & = 20 d^3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 5

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PROBLEM:

Express the area $A$ of an equilateral triangle as a function of its side $x$ .

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SOLUTION:

From the formula of the area of a triangle, $\displaystyle A=\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right)$ . Also, we know that the interior angle of an equilateral triangle is 60 degrees, and $\displaystyle \sin\:60^{\circ} =\frac{\sqrt{3}}{2}$ .

\begin{align*} A & =\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right) \\ A & =\frac{1}{2} \cdot x\cdot x\cdot \sin\:60^{\circ} \\ A & =\frac{1}{2}\cdot x^2\cdot \frac{\sqrt{3}}{2} \\ A & =\frac{\sqrt{3}}{4}x^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Category Archives: Engineering Mathematics Blog

Maximum Load of a Member | Stress | Mechanics of Materials | 3rd Edition | Timothy Philpot | Problem P1.1

A stainless steel tube with an outside diameter of 60 mm and a wall thickness of 5 mm is used as a compression member. If the axial normal stress in the member must be limited to 200 MPa, determine the maximum load P that the member can support.

SOLUTION:

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 3

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 2

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 1

PROBLEM:

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 10

PROBLEM:

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 9

PROBLEM:

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 7

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 6

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 5

Express the area $A$ of an equilateral triangle as a function of its side $x$ .

A stainless steel tube with an outside diameter of 60 mm and a wall thickness of 5 mm is used as a compression member. If the axial normal stress in the member must be limited to 200 MPa, determine the maximum load P that the member can support.

SOLUTION:

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right).

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right).

PROBLEM:

Evaluate \displaystyle \lim _{x\to 2}\left(x^2-4x+3\right).

PROBLEM:

If \displaystyle f\left(x\right)=\frac{4}{x+3} and \displaystyle \:g\left(x\right)=x^2-3 , find \displaystyle f\left[g\left(x\right)\right] and \displaystyle g\left[f\left(x\right)\right].

PROBLEM:

If \displaystyle f\left(x\right)=3x^2-4x+1, find \displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0.

If \displaystyle f\left(x\right)=x^2+1, find \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0.

A right circular cylinder, a radius of base x, height y, is inscribed in a right circular cone, radius of base r and a height h. Express y as a function of x (r and h are constants).

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

Express the area A of an equilateral triangle as a function of its side x.

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

Express the area $A$ of an equilateral triangle as a function of its side $x$ .