Introduction for k 12 math:
Applied math is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. The k 12 math help topics are applications of matrices, vector algebra, complex numbers, analytical geometry, differential calculus, differential calculus – 2, integral calculus, differential equations, discrete mathematics, probability distributions. The k 12 applied math help example problems and practice problems are given below.(Source:Wikipidea)
Example Problems for K 12 Math:
Example problem 1:
Show that the points (3, − 1, − 1), (1, 0, − 1) and (5, − 2, − 1) are collinear.
Solution:
The equation of the line passing through the given point (3, − 1, − 1) and (1, 0, − 1) is
` (x - 3)/2` = `(y + 1)/- 1` = `(z + 1)/0` = λ (say)
Any point on the line is of the form (2λ + 3, − λ − 1, − 1)
The point (5, − 2, − 1) is obtained by putting λ = 1.
The third point lies on the same line. Hence the three points are collinear.
Example problem 2:
The number of bacteria disease in a yeast culture grows at a rate which is proportional to the number present. If the population of a colony of yeast bacteria disease triples in 1 hour. Show that the number of bacteria disease at the end of five hours will be 35 times of the population at initial time.
Solution:
Let A be the number of bacteria at any time t
`(dA)/dt` α A ⇒ `(dA)/dt` = kA ⇒ A = cekt
Initially, i.e., when t = 0, assume that A = A0
A0 = ce° = c
A = A0ekt
when t = 1, A = 3 A0 ⇒ 3A0 = A0 ek ⇒ ek = 3
When t = 5, A = A0 e5k = A0 (ek)5 = 35. A0
The number of bacteria at the end of 5 hours will be 35 times of the number of bacteria at initial time
Example problem 3:
Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.
Given:
PB − PA = 6
` sqrt[(x - 5)^2 + y^2]` − `sqrt[(x + 5)^2 + y^2] ` = 6
Simplifying we get − 9y2 + 16x2 = 144
− `y^2 /16` + `x^2/9` = 1 i.e., `x^2/ 9` − `y^2/ 16` = 1 which is a hyperbola.
These example problems are helpful to study of k 12 math. Please express your views of this topic Probability Distributions by commenting on blog.
Practice Problems for K 12 Math:
Practice problem 1:
Find the area of triangle, The vertices's are (3, − 1, 2), (1, − 1, − 3) and (4, − 3, 1)
Answer: `1/2 sqrt(165)`
`Practice problem 2:`
Prove that the triangle formed by the points representing the complex numbers (10 + 8i), (− 2 + 4i) and (− 11 + 31i) on the Argand plane is right angled.
Applied math is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. The k 12 math help topics are applications of matrices, vector algebra, complex numbers, analytical geometry, differential calculus, differential calculus – 2, integral calculus, differential equations, discrete mathematics, probability distributions. The k 12 applied math help example problems and practice problems are given below.(Source:Wikipidea)
Example Problems for K 12 Math:
Example problem 1:
Show that the points (3, − 1, − 1), (1, 0, − 1) and (5, − 2, − 1) are collinear.
Solution:
The equation of the line passing through the given point (3, − 1, − 1) and (1, 0, − 1) is
` (x - 3)/2` = `(y + 1)/- 1` = `(z + 1)/0` = λ (say)
Any point on the line is of the form (2λ + 3, − λ − 1, − 1)
The point (5, − 2, − 1) is obtained by putting λ = 1.
The third point lies on the same line. Hence the three points are collinear.
Example problem 2:
The number of bacteria disease in a yeast culture grows at a rate which is proportional to the number present. If the population of a colony of yeast bacteria disease triples in 1 hour. Show that the number of bacteria disease at the end of five hours will be 35 times of the population at initial time.
Solution:
Let A be the number of bacteria at any time t
`(dA)/dt` α A ⇒ `(dA)/dt` = kA ⇒ A = cekt
Initially, i.e., when t = 0, assume that A = A0
A0 = ce° = c
A = A0ekt
when t = 1, A = 3 A0 ⇒ 3A0 = A0 ek ⇒ ek = 3
When t = 5, A = A0 e5k = A0 (ek)5 = 35. A0
The number of bacteria at the end of 5 hours will be 35 times of the number of bacteria at initial time
Example problem 3:
Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.
Given:
PB − PA = 6
` sqrt[(x - 5)^2 + y^2]` − `sqrt[(x + 5)^2 + y^2] ` = 6
Simplifying we get − 9y2 + 16x2 = 144
− `y^2 /16` + `x^2/9` = 1 i.e., `x^2/ 9` − `y^2/ 16` = 1 which is a hyperbola.
These example problems are helpful to study of k 12 math. Please express your views of this topic Probability Distributions by commenting on blog.
Practice Problems for K 12 Math:
Practice problem 1:
Find the area of triangle, The vertices's are (3, − 1, 2), (1, − 1, − 3) and (4, − 3, 1)
Answer: `1/2 sqrt(165)`
`Practice problem 2:`
Prove that the triangle formed by the points representing the complex numbers (10 + 8i), (− 2 + 4i) and (− 11 + 31i) on the Argand plane is right angled.
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