The dimensions of a smaller rectangle are 3 ft. by 9 ft. The dimensions of a larger rectangle are 5 ft. by 11 ft. Find the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle. Options are 11:9, 3:4, 4:3, and 5:3 ...?

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.

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.

Answer:

x=4+-isqrt(5/7

Step-by-step explanation:

Answer:

the answer would be D

Step-by-step explanation:

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.

Since the value of m is -17/6, equation c has a negative solution.

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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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.

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.

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.

Answer:

The factors of the  7x² - 12x - 4 = 0 are (7x + 2)(x -2) .

Option (b) is correct i.e none of the above .

Step-by-step explanation:

As the expression given in the question.

7x² - 12x - 4 = 0

7x² - 14x + 2x - 4 = 0

7x (x - 2)  + 2(x - 2) = 0

(7x + 2)(x -2) = 0

Thus the factors of the  7x² - 12x - 4 = 0 are (7x + 2)(x -2) .

Option (b) is correct i.e none of the above .

Answer:

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Step-by-step explanation:

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

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

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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

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.