A Problem on Volume of a Cube/Geometrical Figure Cube

All the geometrical construction and geometrical figures like cube, cuboid, prism etc come under construction math and construction geometry where you will get more information about these geometrical figures and there attributes.

Topic : Volume of a cube

A Cube is a Geometrical Figure with dimension of equal sides, here is one such example on Cube, to Find length of the side of cube, given volume of cube using indice laws.

Question : The volume of a cube is given by V = s³, where s is the length of a side. Find the length of a side of a cube if the volume is 700 in³.Round the answer to three decimal places.

Solution :

Given V = 700 in³

Also, volume of a cube is given by

V = s³

=> 700 = s³

=> s³ = 700

=> s = cuberoot(700)

=> s = 8.697

Length of a side of cube = 8.697 inches.

Word problem on Function and gas concentration as its dependencies

Functions are denoted as shown in the figure where there will be dependencies of one variable value to another.

Topic : Functions and variables values.

Variables, co-efficient are the terms found in functions.

Problem : The table lists future concentrations of a certain green house gas in parts per billion (ppb), If current trends continue.

a) Let x = 0 correspond to 2010 and x = 20 correspond to 2030. Find C and a so that f(x) = Ca^x models the data.

b) Estimate the gas concentration in 2021.

Year ------ Gas(ppb)
2010 ------ 0.74 (when x = 0)
2015 ------ 0.86
2020 ------ 0.99
2025 ------ 1.15
2030 ------ 1.34 (when x = 20)

Solution :

a) We have function ,
f(x) = Ca^x

x = 0 ; f(0) = Ca⁰ = C*1 = C

So f(0) = 0.74 ; C = 0.74

and x = 20 ; f(x) = C a²⁰ = 0.74 a²⁰

f(20) = 1.34 (from the table)

So 1.34 = 0.74 a²⁰ ; a²⁰ = 1.34/0.74 = 1.8108

So a = 1.0301


b) To find gas concentration at the year 2021, we have to find f(21)

f(21) = Ca²¹ = 0.74 (1.0301)²¹

= 0.74 * 1.8641

= 1.37943

So f(21) approximately 1.38

Therefore gas concentration at 2021 will be 1.38

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Question on Integrating Factor Method for Solving Differential Equation

Topic : Integrating Factor

Question : Explain Integrating Factor Method for Solving Differential Equation

Solution :
Suppose we have this equation to solve:
dy + y = x
dx
In this case we cannot separate the variables and introducing a new variable doesn't help either. So we need to use an entirely different method, known as Integrating Factor Method.
Here's the typical equation for which we can use this method:
dy + yP(x) = Q(x)
dx
Here the functions P(x) and Q(x) can be any functions of x, in the example above they were 1 and x respectively.
Step One: calculate the integral of the function P(x).
Step Two: the integrating factor, which we'll call IF, is defined as the exponential of this, i.e. it's defined by :
Integrating Factor = IF = e^(∫P(x) dx)
Step Three: the solution to the equation is given by:
y(x) = 1/IF ∫Q(x) IF dx
Now let us understand this using an example-
dy + y = x
dx
So P(x)=1 and Q(x)=x. The integrating factor is therefore ex, since the integral of 1 is x. (No need for a constant of integration, that's because if you do put one in it will cancel out later on anyway)
So, we can write down the solution y(x) of the equation using the "mysterious" formula given in Step Three:
y(x) = 1/IF ∫Q(x) IF dx
In this case that's:
y(x) = 1/e^x ∫xe^x dx
We can do that integral using integration by part. The result is:
y(x) = e^(-x) (xe^x - e^x + c)
which implies to
y(x) = x - 1 + ce^(-x)

Problem to Solve Simultaneous Equations by Substitution Method

Topic : Substitution Method

Problem : Solve using Substitution Method.

y - 2x = - 6

2y - x = 5

Solution :

Question to Find Factors of a Function

Topic : Functions

Question : Given f(x) = 2x³ - 5x² + 12x - 5 Find all zeros, if

p : ± 1, ± 5

q : ± 1, ± 2

p/q = ± 1/2 , ± 5/2

Solution :

Question on Linear Equation

Topic : Linear Equation

Question : Sue wants to buy a new printer that costs $189. She has $125 saved. She has a job that pays $8 an hour. How many hours must she work to earn enough to buy the printer?

Solution :
The cost of the printer is $189.
She has saved $125.
Also, she is paid $8 an hour to do a job.
We need to find the number of hours she works to buy the printer.
Let us assume “x” to be the number of hours she works to buy the printer.
8x + 125 =189
Now, subtract 125 on either side of the equation.
8x + 125 – 125 = 189 – 125
8x = 64
Divide by 8 on either side of the equation.
8x/8 = 64/8
x = 8
Therefore, the number of hours she works to buy the printer is 8.

Solved variable Saparable form of Differential Equation

Topic : Variable Saparable Form

Question : Solve dy/dx = (x-1)/(y+2)

Solution :
dy/dx = (x-1)/(y+2)
(y+2) dy = (x-1) dx
∫(y+2) dy = ∫(x-1) dx
y²/2 + 2y = x²/2 - x + c1 (c1 is the constant)