500/100 in simplest form
If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then ...?
solve the differential equation:
(1-2x^2-2y)dy/dx=4x^3+4xy
Answer:
[tex]u=-x^4-2x^2y+y-y^2[/tex]
Step-by-step explanation:
We are given that
[tex](1-2x^2-2y)\frac{dy}{dx}=4x^3+4xy[/tex]
We have to solve the given differential equation
[tex](1-2x^2-2y)dy=(4x^3+4xy)dx[/tex]
[tex](1-2x^2-2y)dy-(4x^3+4xy)dx=0[/tex]
Compare with [tex]Mdx+ndy=0[/tex]
Then, we get [tex]M=-(4x^3+4xy),N=(1-2x^2-2y)[/tex]
Exact differential equation
[tex]M_y=N_x[/tex]
[tex]M_y=-4x[/tex]
[tex]N_x=-4x[/tex]
[tex]M_y=N_x[/tex]
Hence, the differential equation is an exact differential equation.
Solution of exact differential is given by
[tex]u=\int M(x,y)dx+K(y)[/tex] where K(y) is a function of y.
[tex]u=\int -(4x^3+4xy) dx+k(y)[/tex] y treated as constant
[tex]u=-x^4-2x^2y+k(y)[/tex]
[tex]u_y=N[/tex]
[tex]-2x^2+k'(y)=1-2x^2-2y[/tex]
[tex]K'(y)=1-2y[/tex]
[tex]k(y)=y-y^2[/tex]
Substitute the value then we get
Then, [tex]u=-x^4-2x^2y+y-y^2[/tex]
Final answer:
The given differential equation is exact, and by integrating the respective terms and finding the function H(y), the solution for z(x, y) for the differential equation is determined to be
-x⁴ + y - y² + C, where C is a constant.
Explanation:
To solve the differential equation (1-2x² - 2y)dy/dx = 4x³ + 4xy, let's rearrange the equation in the form M(x, y)dy + N(x, y)dx = 0, which is the standard form for a first-order differential equation. We can rewrite our equation as (-4x³ - 4xy)dx + (1-2x² - 2y)dy = 0.
Now, we check if the equation is exact, that is, if ∂M/∂y = ∂N/∂x. If it is exact, there exists a function z(x, y) such that dz = M dy + N dx.
In this case, ∂M/∂y = -4x, and ∂N/∂x = -4x. Since the partial derivatives are equal, the differential is exact, meaning there exists some function z(x, y) such that dz = Ndx + Mdy. To find z(x, y), we integrate N with respect to x and M with respect to y and combine the resulting functions, making sure to include the functions of the other variable that may arise from the partial integration.
For N(x, y), we integrate -4x³dx to get -x⁴. For M(y), we integrate (1-2y)dy to get y - y². Thus z(x, y) = -x⁴ + y - y² + H(y), where H(y) is a function of y.
To find H(y), differentiate z with respect to y and equate it to M: dz/dy = 1 - 2y + H'(y) = 1 - 2y, which implies that H'(y) = 0, hence H(y) is a constant. Therefore, the function that satisfies the differential equation is z(x, y) = -x⁴ + y - y² + C, where C is a constant.
Henry is saving money for college. He earns $210 each week working part time after school and the weekends. Henry currently has $2,240 in savings. His parents put $100 in his savings account each week and he saves one-third of his paycheck each week. Which expression represents the situation. (n represents the number of weeks)
Answer:
[tex]f(n)=2240+(\frac{170}{Week}).n[/tex]
Step-by-step explanation:
In order to make an expression that represents the situation we need to make a function that represents Henry's savings.
This will be a function ''f(n)'' because it will depend of the variable ''n'' which is the number of weeks.
Let's start making the function by reading the problem.
''He earns $210 each week working part time after school and the weekends''
We can write :
[tex]f(n)=(\frac{210}{Week}).n[/tex]
Where ''210'' is actually $210
''Henry currently has $2240 in savings'' ⇒ We need to add this amount of money to the function.
[tex]f(n)=(\frac{210}{Week}).n+2240[/tex]
''His parents put $100 in his savings account each week and he saves one-third of his paycheck each week'' ⇒ We need to add ($100).n to the expression due to his parents and multiply by [tex]\frac{1}{3}[/tex] the expression that represents its paycheck ⇒
[tex]f(n)=(\frac{1}{3}.\frac{210}{Week}).n+2240+(\frac{100}{Week}).n[/tex]
Now, if we work with the expression :
[tex]f(n)=(\frac{70}{Week}).n+2240+(\frac{100}{Week}).n[/tex]
[tex]f(n)=2240+(\frac{170}{Week}).n[/tex]
Where the units of ''2240'' and ''170'' are $.
That is the final expression which represents the situation.
An arhitect desings a rectangular flower garden such that the width is exactly two-thirds of the length. If 280 feet of antique picket fencing are to be used to enclose the garden, find the dimensions of the garden.
i really need help with my homework.
factorise 10x+25
Which graph represents the solution set for the system 2x + 5y ≤ 9 and 3x + 5y ≤ 9?
Answer:
The graph representing the solution is given below.
Step-by-step explanation:
We are given the system of inequality is,
[tex]2x + 5y\leq 9[/tex]
[tex]3x + 5y \leq 9[/tex]
Zero Test states that,
'After substituting the point (0,0) in the inequalities, if the result is true, then the solution region id towards the origin. If the result is false, the solution region is away from the origin'.
So, upon substituting (0,0) in the given inequalities, we get,
[tex]2x + 5y\leq 9[/tex] implies 0≤ 9, which is true.
[tex]3x + 5y \leq 9[/tex] implies 0≤ 9, which is true.
Thus, the solution region for both the inequalities is towards the origin.
Hence, upon plotting, the graph representing the solution set is given below.
A right triangle is removed from a rectangle to create the shaded region shown below. Find the area of the shaded region. Be sure to include the correct unit in your answer.
Answer:
The area of shaded region is 45 square units.
Step-by-step explanation:
The length of the rectangle is 8 and the breadth is 6.
Area of rectangle is
[tex]A_R=l\times b[/tex]
[tex]A_R=8\times 6=48[/tex]
The area of rectangle is 48 square units.
Since the opposite sides of a rectangle are equal, therefore the legs of the right angled triangle are
[tex]l_1=8-5=3[/tex]
[tex]l_2=6-4=2[/tex]
The area of right angled triangle is
[tex]A_T=\frac{1}{2}\times l_1\times l_2[/tex]
[tex]A_T=\frac{1}{2}\times 3\times 2=3[/tex]
The area of triangle is 3 square units.
The area of shaded region is
[tex]A=A_R-A_L=48-3=45[/tex]
Therefore the area of shaded region is 45 square units.
Find the roots of the equation below. 7x2 + 3 = 8x
A) -4/7 (+ or -) (sqrt 5)/7
B) 4/7 (+ or -) (sqrt 5)/7
C) -4/7 (+ or -) i(sqrt 5)/7
D) 4/7 (+ or -) i(sqrt 5)/7
Answer:
x=4+-isqrt(5/7
Step-by-step explanation:
what line is parallel to x-3y=24
What is 8 1/4 devided by 1/2?
A box with a square base of side a is three times higher than its width. Express the volume V of the box as a function of a.
V(a) = ?
The volume V of a box with a square base of side a and a height that is three times its width is expressed as the function V(a) = 3a³.
To express the volume V of a box as a function of its base side length a, with the height being three times its width, we can use the formula for the volume of a rectangular prism, which is the product of its length, width, and height. As the base is a square with side a, both the length and the width of the base are a. Given that the box's height is three times its width (or length, since it is a square), the height will be 3a.
Therefore, the volume V of the box as a function of a is V(a) = a² × (3a) = 3a³.
The golden ratio or golden mean is represented as (1 + √5) : 2.
What is its decimal value to the nearest thousandth? (Hint: Use a calculator to evaluate (1+√5) divided by 2.)
The golden ratio or golden mean is approximately 1.618 when calculated to the nearest thousandth using a calculator to evaluate (1 + √5) / 2.
The golden ratio or golden mean is a number often encountered in mathematics and art, and it is commonly represented by the Greek letter phi (φ). It can be expressed mathematically as (1 + √5) / 2. To find its decimal value to the nearest thousandth, we use a calculator to perform the operation.
First, calculate the square root of 5, then add 1 to the result. After this, divide the sum by 2. This will give us the golden ratio:
√5 ≈ 2.236
1 + √5 ≈ 3.236
(1 + √5) / 2 ≈ 1.618
So, to the nearest thousandth, the decimal value of the golden ratio is 1.618.
The derivative of f(x)=(x^4/3)-(x^5/5) attains its maximum at x= ? ...?
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
Further explanation:
Given:
The function is [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Explanation:
The given function is [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Differentiate the above equation with respect to x.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right) &= \frac{d}{{dx}}\left( {\frac{{{x^4}}}{3} - \frac{{{x^5}}}{5}} \right)\\&= \frac{{4{x^3}}}{3} - \frac{{5{x^4}}}{5}\\&= \frac{{4{x^3}}}{3} - {x^4}\\\end{aligned}[/tex]
Again differentiate with respect to x.
[tex]\begin{aligned}\frac{{{d^2}}}{{d{x^2}}}f\left( x \right) &= \frac{{{d^2}}}{{d{x^2}}}\left( {\frac{{4{x^3}}}{3} - {x^4}} \right)\\&=\frac{{3 \times 4{x^2}}}{3} - 4{x^3}\\&= 4{x^2} - 4{x^3}\\\end{aligned}[/tex]
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right)&= 0\\\frac{{4{x^3}}}{3} - {x^4}&= 0\\\frac{{4{x^3}}}{3} &= {x^4}\\\frac{4}{3}&= \frac{{{x^4}}}{{{x^3}}}\\\frac{4}{3}&= x\\\end{aligned}[/tex]
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, attains, maximum, value of x, function, differentiate, minimum value.
whats the difference of 1cm and 1cm2
Final answer:
1 cm is a unit of length equal to [tex]10^{-2}[/tex], meter while [tex]1 cm^2[/tex]is a unit of area equal to 10^{-4} square meters.
Explanation:
The difference between 1 cm and 1 cm2 is that 1 cm is a measure of length, while 1 cm2 is a measure of area. To clarify, 1 cm is equivalent to 10-2 meters (or 0.01 meters), and it represents a one-dimensional measurement. In contrast, 1 cm2 is the area of a square with 1 cm long sides. When converting to square meters (m2), remember that the conversion factor for area is the square of the conversion factor for length. Therefore, 1 cm2 equals (10-2 m)2 or 10-4 m2.
Which expression is equal to the number of grams (g) in 2.43 kilograms (kg)?
Answer:
the answer would be D
Step-by-step explanation:
Before beginning voice lessons, Justin already knew how to sing 1 piece, and he expects to learn 2 new pieces during each week of lessons. Write an equation that shows the relationship between the number of weeks x and the number of pieces learned y. Then Graph.
U have 5 pens. u get 5 more pens. how many pens do u have now
determine the exact value of the trig ratios:
cos(13pi/4)
cot(11pi/2)
sec(5pi/3)
This isn't the only trick you'll need, but it will help you get these expressions in terms you'll be able to handle more easily. For example, for the cot example above: cot(-pi/2) = cos(-pi/2)/sin(-pi/2) = 0/-1 = 0
The exact values of the trig ratios are sqrt(2)/2, 0, and 2, for cos(13pi/4), cot(11pi/2), and sec(5pi/3) respectively.
To determine the exact value of the trig ratios, we can use the unit circle and trigonometric identities.
cos(13pi/4):
The angle 13pi/4 represents a full circle plus one revolution, so the cosine value is the same as cos(pi/4).
cos(pi/4) = sqrt(2)/2
cot(11pi/2):
The angle 11pi/2 represents 5 full circles plus a half revolution, so the cotangent value is the same as cot(pi/2).
cot(pi/2) = 0
sec(5pi/3):
The angle 5pi/3 represents 2 full circles plus two-thirds of a revolution, so the secant value is the same as sec(pi/3).
sec(pi/3) = 2
write 2/5 as a decimal
If the hypotenuse and an acute angle of one right triangle are equal to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent by the HA theorem. If a leg and an acute angle of one right triangle are equal to the corresponding parts of another right triangle, then the triangles are congruent by the _______ theorem.
AL
LA
HL
b?
Trevor solved the system of equations below. What mistake did he make in his work?
2x + y = 5
x − 2y = 10
y = 5 − 2x
x − 2(5 − 2x) = 10
x − 10 + 4x = 10
5x − 10 = 10
5x = 0
x = 0
2(0) + y = 5
y = 5
He should have substituted 5 + 2x
He combined like terms incorrectly, it should have been 4x instead of 5x
He subtracted 10 from the right side instead of adding 10 to the right side
He made no mistake ...?
Answer:
Give the guy above me Brainlyist
Step-by-step explanation:
A laptop computer is purchased for $1550 . after each year, the resale value decreases by 25% . what will the resale value be after 3 years?
(AB)^2 + (BC)^2 = (AC)^2
BC = in.
What did the people learn about the banks during this fireside chat?
how do you find volume of a cube
Determine if the ordered pair is a solution of the equation.
Is (2,4) a solution of y = 10 -3x?
Question 3 options:
True
False
Write the quadratic function in vertex form.
y = x2 - 2x + 5
Answer:
[tex]y=(x-1)^{2}+4[/tex]
Step-by-step explanation:
To write this quadratic function in vertex form, which is the explicit form of the parabola, we have to complete the square in the expression.
First, we have to take the coefficient of the linear term and find the squared power of its half:
[tex](\frac{b}{2} )^{2}=(\frac{2}{2} )^{2}=1[/tex]
Then, we add and subtract this number in the quadratic expression:
[tex]y=x^{2}-2x+5+1-1[/tex]
Now, we use the three terms that can be factorize as the squared power of a binomial expression:
[tex]y=(x^{2}-2x+1)+5-1[/tex]
Then, we find the square root of the first term and third term, and we form the squared power:
[tex]y=(x-1)^{2}+4[/tex]
Now, this vertex form is explicit, because it says from the beginning what's the coordinates of the vertex, which is: [tex](1;4)[/tex], as minimum, because the parabola is concave up.
In vertex form, the quadratic equation is written as;
⇒ y = (x - 1)² + 4
What is Quadratic equation?An algebraic equation with the second degree of the variable is called an Quadratic equation.
We have to given that;
The quadratic equation is,
⇒ y = x² - 2x + 5
Now, We can write in vertex form as;
⇒ y = x² - 2x + 5
⇒ y = x² - 2x + 1 + 4
⇒ y = (x - 1)² + 4
Learn more about the quadratic equation visit:
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12 tenths plus 17 hundredths
Which of the following equations will have a negative solution?
A.m - 4 2/3 = 7 1/2
B.m - 71/2 = 4 2/3
C.7 1/2 + m = 4 2/3
D.-7 1/2 + m = 4 2/3
Since the value of m is -17/6, equation c has a negative solution.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that,
7 1/2 + m = 4 2/3
We have to apply the arithmetic operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.
15/2+m=14/3
Convert the fraction value and rearrange the equation as follows,
m=14/3-15/2
m=-17/6
Thus, the value of m is -17/6, equation c has a negative solution.
Learn more about the equation here,
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