5/8 of the 64 musicians in a music contest are guitarist. Some of the guitarist play jazz solos, and the rest play classical solos. The ratio of the number of guitarist playing jazz solos to the total number of guitarist in the contest is 1:4. How many guitarist play classical solos in the contest?

In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the coefficient of proportionality or proportionality constant.

The equation of the parabola is y = ax² = (-1/2)x² where a = -2 and b = -8 are the required values of a and b.

The first derivative of y = ax² that represents the slope of the tangent line to the curve of y = ax² .  

Here, dy/dx = 2ax.  

When x = 2,

dy/dx = 2a(2) = 4a.

The point of tangency is (2,y), where y = a(2)², or y=4a;

thus, the point of tangency is; (2, 4a).  

The equation of the tangent line to y=ax² at (2,4a)

Now differentiating y=ax² with respect to x,

   dy/dx = 2ax  

dy/dx (at 2,4a) = 2a(2) = 4a

The line 2x + y = b is tangent to y = ax² at (2,4a).

The slope of 2x + y = b can be found by solving 2x + y = b for y:

y = b - 2x        

Slope m = -2

Thus, dy/dx = 4a = - 2, and thus a = -2/4, or a = -1/2.  All we have to do now is to find the value of b.  

We know that 2x + y = b, then if x=-2 and y=-8,

2(-2) + [-8] = b = -4 - 8 = -12

Thus, the equation of the parabola is y = ax² = (-1/2)x² where a = -2 and b = -8 are the required values of a and b.

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Answer:

The maximum number of students that can attend each meeting is:

40

Step-by-step explanation:

A school conference room can seat a maximum of 83 people.

The principal and two counselors need to meet with the school’s student athletes to discuss eligibility requirements.

Let there be x students,there will be x parents

that means x+x+3≤83

Subtracting both sides by 3,we get

2x≤80

dividing both sides by 2,we get

x≤40

Hence, the maximum number of students that can attend each meeting is:

40

Answer:

[tex](4 \times 1)+(3 \times \frac{1}{100})+(5 \times \frac{1}{1000})[/tex]

Step-by-step explanation:

Given : MARIA WALKED 4.035 KILOMETERS.

To Find :WHAT IS 4.035 WRITTEN IN AS EXPANDED FORM ?

Solution :

Number = 4.035

The numbers after decimals when read from first to last has positions tenth , hundredth , thousandth and so ...

The number before the decimal part has positions ones , tens , hundreds and so on when read from last to first

Now 4 is at ones place

0 is at tenth place

3 is at hundredth place

5 is at thousandth place

So, Expanded form : [tex](4 \times 1)+(3 \times \frac{1}{100})+(5 \times \frac{1}{1000})[/tex]

Final answer:

The volume of the rectangular block of ice is 85.5 cubic feet. This was found by converting the mixed numbers into improper fractions, then multiplying the length, width, and height together using the formula volume = length × width × height.

Explanation:

To find the volume of a rectangular block of ice with the given dimensions, we simply need to multiply the length, width, and height together. The formula to calculate volume is Volume = length × width × height. First, however, we need to convert the mixed numbers into improper fractions so we can multiply them easily.

The length is given as 6 1/3 feet, which can be converted to an improper fraction: (6 × 3) + 1 = 19/3 feet. The height is given as 1 1/2 feet, which is (1 × 2) + 1 = 3/2 feet.

Now, to find the volume, we multiply these dimensions with the width, which is 3 feet.

Volume = (19/3) feet × 3 feet × (3/2) feet
The feet × feet × feet will give us cubic feet.

Multiplying these together:
Volume = (19 × 3 × 3) / (3 × 2) cubic feet
Volume = 171/2 cubic feet or 85.5 cubic feet

Thus, the volume of the block of ice is 85.5 cubic feet.

To ascertain if a vector field is conservative or not, you need to calculate the curl of the field or integrate over the components of the vector field. If the curl is zero, it's conservative. If the curl isn't zero or an integral doesn't exist, the field is not conservative.

To determine if the vector field f(x, y, z) = ye−xi + e−xj + 2zk is conservative, we need to find if there exists a function f such that f is the gradient (denoted by ∇) of f. This can be done by checking if the cross product of the vector field is equal to zero, which signifies that the field is conservative.

First, we calculate the curl (∇ x F) of the vector field, which gives us the derivatives of the field components. If the curl is zero, then the vector field is conservative. If it is not zero, this indicates that the vector field is non-conservative.

Alternatively, we can integrate over the components of the vector field to try and find a potential function. If an integral exists, then we can say that the vector field is conservative.

However, if it fails these conditions, then the vector field is not conservative and the function f does not exist for it (dne). Thus, in the case where the vector field is not conservative, enter 'dne'.

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Answer:

Customers older than 45 years are 11 in number.

Step-by-step explanation:The age of store customers is represented by the stem ad leaf plot.The stem represents the tens digit while leaf denotes the unit digit.The question is asking us to find the number of customers who are older than 45 so 45 is not considered.

The age of customers more than 45 are:

48,50,50,51,55,56,62,64,65,65,73.There are 11 customers in all .

The coefficient of the [tex]\(x^4\) term in \((x + 3)^{12}\)[/tex] is 495, calculated using binomial coefficients.

To find the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex], you can use the binomial theorem. According to the binomial theorem, the expansion of [tex]v[/tex] is given by:

[tex]\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\][/tex]

Where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, equal to [tex]\(n\) choose \(k\)[/tex], which is defined as:

[tex]\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\][/tex]

In this case, [tex]\(n = 12\) and \(y = 3\).[/tex] We're interested in the term where the exponent of [tex]\(x\) is 4, so \(n - k = 4\) or \(k = 12 - 4 = 8\).[/tex]Thus, we need to find the coefficient when [tex]\(k = 8\)[/tex]. So, the coefficient of the [tex]\(x^4\)[/tex] term is:

[tex]\[\binom{12}{8} = \frac{12!}{8!(12-8)!}\][/tex]

Calculating this:

[tex]\[\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495\][/tex]

So, the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex] is 495.

Answer : The fraction of pounds of fruits over the total pounds of fruit and vegetables is, [tex]\frac{1005}{2669}[/tex]

Step-by-step explanation :

As we are given that:

Total amount of vegetables = 208 pounds

Total amount of fruits = [tex]125\frac{5}{8}\text{ pounds}=\frac{1005}{8}\text{ pounds}[/tex]

Thus, the total amount of fruits and vegetables will be:

[tex]208+125\frac{5}{8}\\\\=208+\frac{1005}{8}\\\\=\frac{1664+1005}{8}\\\\=\frac{2669}{8}[/tex]

Now we have to calculate the fraction of pounds of fruits over the total pounds of fruit and vegetables :

[tex]\frac{\text{Pounds of fruits }}{\text{ total pounds of fruits and vegetables}}\\\\=\frac{(\frac{1005}{8})}{(\frac{2669}{8})}\\\\=\frac{1005}{8}\times \frac{8}{2669}\\\\=\frac{1005}{2669}[/tex]

Thus, the fraction of pounds of fruits over the total pounds of fruit and vegetables is, [tex]\frac{1005}{2669}[/tex]

The given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex]  is indeed linear, and it can be expressed as f(x) = x in the standard linear form.

Let's break down the analysis of the given function[tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex].

1. Initial Expression:

 [tex]\[ f(x) = \frac{5}{5} \cdot x \][/tex]

2. Simplify the Fraction:

 [tex]\[ \frac{5}{5} \][/tex] simplifies to 1, so the expression becomes:

[tex]\[ f(x) = 1 \cdot x \][/tex]

3. Multiplication by 1:

  Multiplying any value by 1 does not change the value, so the expression further simplifies to:

  f(x) = x

4. Linear Form:

  The function is now in the form f(x) = ax + b with a = 1 and b = 0:

[tex]\[ f(x) = 1 \cdot x + 0 \][/tex]

Therefore, the given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex]  is indeed linear, and it can be expressed as f(x) = x in the standard linear form.

Complete Question: Determine whether the given function is linear. if the function is linear, express the function in the form f(x) = ax + b. (if the function is not linear, enter not linear.)

f(x) = 5 / 5 x

A negative number is raised to an odd exponent. The result is always negative.

What is mean by Odd exponent?

An odd power of a number is a number of the form for the integer and a positive odd integer.

The first few odd powers are 1, 3, 5, 7, .........

Given that;

The expression is;

A negative number is raised to an odd exponent.

Now, To prove this statement that ''A negative number is raised to an odd exponent. The result is always negative.''

Let an example for an odd exponent as;

f (x) = (- 4)³

Here the power is 3 which is odd.

This gives;

f (x) = (- 4)³

f (x) = - 64

Which is negative function.

Hence, A negative number is raised to an odd exponent is always negative.

Therefore,

A negative number is raised to an odd exponent. The result is always negative.

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Answer:

Yes

Step-by-step explanation:

ye mom ye mom lolololol

Answer: sin x° = 3/z( answer

Because tan is opposite/adjacent,

Cos is adjacent/hypotenuse and sin is opposite/hypotenuse the information to find sin is given. You simply take the opposite (3) and put it over the hypotenuse (z)

sin x°= 3/z

The volume of a cuboid is given as length × width × height thus the volume of the paperback book will be 882 cm².

Volume is the scalar quantity of any object that specified occupied space in 3D.

Volume has units of cube example meter³,cm³, etc.

Given a paperback book shape as cuboid

Length(tall) = 21 cm

Width(wide) = 12cm

Height (thick) = 3.5 cm

The volume of the cuboid is given as

Volume = length × height × width.

Volume = 21 × 12 × 3.5

Volume = 882 cm²

Hence" The volume of a cuboid is given as length × width × height thus the volume of the paperback book will be 882 cm²".

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Answer:

[tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Functions

Calculus

Differentiation

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Taylor Polynomials

Step-by-step explanation:

*Note: I will not be showing the work for derivatives as it is relatively straightforward. If you request for me to show that portion, please leave a comment so I can add it. I will also not show work for elementary calculations.

Step 1: Define

Identify

f(x) = √(x² + 8)

Center: x = 1

n = 2

Step 2: Differentiate

Step 3: Evaluate

Step 4: Write Taylor Polynomial

Topic: AP Calculus BC (Calculus I + II)  

Unit: Taylor Polynomials and Approximations  

Book: College Calculus 10e

The second degree Taylor polynomial for the function [tex]f(x) = sqrt(x^2+8)[/tex] at x=1 is [tex]T(x) = 3 + (x-1) - 1/32(x-1)^2[/tex].

To find the second degree Taylor polynomial for [tex]f(x) = sqrt(x^2+8)[/tex] at the number x=1, we begin by calculating the necessary derivatives and evaluating them at x=1. The Taylor polynomial of degree n at x=a is given by:

[tex]T(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n[/tex].

In this case, we need to find the first and second derivatives:

[tex]f'(x) = \frac{1}{2}(x^2+8)^{-1/2} · 2x[/tex]

[tex]f''(x) = \frac{1}{2} · (-1/2)(x^2+8)^{-3/2} · 2x^2 + \frac{1}{2}(x^2+8)^{-1/2}[/tex]

Then we evaluate f(x), f'(x), and f''(x) at x=1:

[tex]f(1) = sqrt(1^2+8) = sqrt9 = 3[/tex]

[tex]f'(1) = \frac{1}{2}(1^2+8)^{-1/2} · 2 · 1 = 1[/tex]

[tex]f''(1) = \frac{1}{2} · (-1/2)(1^2+8)^{-3/2} · 2 · 1^2 + \frac{1}{2}(1^2+8)^{-1/2} = -\frac{1}{16}[/tex]

Thus, the second degree Taylor polynomial at x=1 is:

[tex]T(x) = 3 + (x-1) - \frac{1}{32}(x-1)^2[/tex].