The ph of water samples from a speciﬁc lake is a random variable y with probability density function given by f (y) = $ (3/8)(7 − y) 2 , 5 ≤ y ≤ 7, 0, elsewhere. a find e(y ) and v(y ). b find an interval shorter than (5, 7) in which at least three-fourths of the ph measurements must lie. c would you expect to see a ph measurement below 5.5 very often? why?

Answers

Answer 1

Given that the ph of water samples from a speciﬁc lake is a random variable y with probability density function given by

[tex]f(y)= \left \{ {{ \frac{3}{8}(7-y)^2 \ \ \ 5\leq y\leq7 } \atop {0 \ \ \ \ \ \ \ elsewhere}} \right. [/tex]

Part A:

[tex]E(y)= \int\limits^\infty_{-\infty} {yf(y)} \, dy \\ \\ = \int\limits^7_5 {y \left(\frac{3}{8}\right)(7-y)^2} \, dy\\ \\ = \frac{3}{8}\int\limits^7_5 (49y-14y^2+y^3)dy \\ \\ = \frac{3}{8} \left[ \frac{49}{2} y^2- \frac{14}{3} y^3+ \frac{1}{4} y^4\right]^7_5 \\ \\ = \frac{3}{8} [(1,200.5-1,600.67+600.25)-(612.5-583.33+156.25)] \\ \\ = \frac{3}{8} (200.08-185.42)= \frac{3}{8} (14.66)=5.5[/tex]

Part B:

[tex]E(y^2)= \int\limits^\infty_{-\infty} {y^2f(y)} \, dy \\ \\ = \int\limits^7_5 {y^2 \left(\frac{3}{8}\right)(7-y)^2} \, dy\\ \\ = \frac{3}{8}\int\limits^7_5 (49y^2-14y^3+y^4)dy \\ \\ = \frac{3}{8} \left[ \frac{49}{3} y^3- \frac{14}{4} y^4+ \frac{1}{5} y^5\right]^7_5 \\ \\ = \frac{3}{8} [(5,602.33-8,403.5+3,361.4)-(2,041.67-2,187.5+625)] \\ \\ = \frac{3}{8} (560.23-479.17)= \frac{3}{8} (81.06)=30.4[/tex]

[tex]V(y)=E(y^2)-[E(y)]^2=30.4-(5.5)^2=30.4-30.25=0.15[/tex]

Part C:

Let the required interval be (5, b), then

[tex]P(5\leq y\leq b)= \frac{3}{4} \\ \\ \Rightarrow \int\limits^b_5 {\left(\frac{3}{8}\right)(7-y)^2} \, dy = \frac{3}{4} \\ \\ \Rightarrow\int\limits^b_5 {(49-14y+y^2)} \, dy=2 \\ \\ \Rightarrow \left[49y-7y^2+ \frac{1}{3}y^3\right]^b_5=2 \\ \\ \Rightarrow (49b-7b^2+\frac{1}{3} b^3)-(245-175+41.67)=2 \\ \\ \Rightarrow \frac{1}{3} b^3-7b^2+49b-113.67=0 \\ \\ \Rightarrow b=5.74 [/tex]

Therefore, an interval shorter than (5, 7) in which at least three-fourths of the ph measurements must lie is (5, 5.74).

Part D:

[tex]P(5\ \textless \ Y\ \textless \ 5.5)=\int\limits^{5.5}_5 {\left(\frac{3}{8}\right)(7-y)^2} \, dy \\ \\ = \frac{3}{8}\int\limits^{5.5}_5 {(49-14y+y^2)} \, dy=\frac{3}{8}\left[49y-7y^2+ \frac{1}{3}y^3\right]^{5.5}_5\\ \\ \frac{3}{8}[(269.5-211.75+55.46)-(245-175+41.67)]=\frac{3}{8}[113.21-111.67] \\ \\ \frac{3}{8}(1.54)=0.5775[/tex]

Since, the probability that a ph measurement is below 5.5 significant (i.e. 57.75%), we would expect to see a ph measurement below 5.5 very often.

Related Questions

Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = ye−xi + e−xj + 2zk

Answers

A three-dimensional vector field is conservative if it is also irrotational, i.e. its curl is [tex]\mathbf 0[/tex]. We have

[tex]\nabla\times\mathbf f(x,y,z)=-2e^{-x}\,\mathbf k[/tex]

so this vector field is not conservative.

- - -

Another way of determining the same result: We want to find a scalar function [tex]f(x,y,z)[/tex] such that its gradient is equal to the given vector field, [tex]\mathbf f(x,y,z)[/tex]:

[tex]\nabla f(x,y,z)=\mathbf f(x,y,z)[/tex]

For this to happen, we need to satisfy

[tex]\begin{cases}f_x=ye^{-x}\\f_y=e^{-x}\\f_z=2z\end{cases}[/tex]

From the first equation, integrating with respect to [tex]x[/tex] yields

[tex]f_x=ye^{-x}\implies f(x,y,z)=-ye^{-x}+g(y,z)[/tex]

Note that [tex]g[/tex] *must* be a function of [tex]y,z[/tex] only.

Now differentiate with respect to [tex]y[/tex] and we have

[tex]f_y=-e^{-x}+g_y=e^{-x}\implies g_y=2e^{-x}\implies g(y,z)=2ye^{-x}+\cdots[/tex]

but this contradicts the assumption that [tex]g(y,z)[/tex] is independent of [tex]x[/tex]. So, the scalar potential function does not exist, and therefore the vector field is not conservative.

Final answer:

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.

Explanation:

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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Factor out the coefficient of the variable. The expression 2.2x +4.4 factored is

Answers

2.2(x+2) is the answer

The Perez family has a rectangular fish tank how much water will the tank hold if it's length is 3 feet, its width is 2 feet and its height is 3 feet

A) NONE
B) 18 cubic feet
C) 3 cubic feet
D) 6 cubic feet
E) 9 cubic feet

Answers

V = 3 x 2 x 3 = 18
answer
B) 18 cubic feet

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. If each student must bring a parent with them, what is the maximum number of students that can attend each meeting?

Answers

if you have only 83 seats, and 3 of which are being occupied by the principal and two counselors, (3 people that will always be at the meetings) you only have 80 seats for the students and the parents. If each student brings only 1 parent with them, divide 80 by 2 ( 2 is a pair of student and parent ) you will have 40 seats. You can only invite 40 Students

Answer:

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

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:

if 2 pounds of strawberries cost 4.50 how much would 3 pounds cost

Answers

6.75$ this my best answer

[tex]\bf \begin{array}{ccll} lbs&\$\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 2&4.50\\ 3&x \end{array}\implies \cfrac{2}{3}=\cfrac{4.50}{x}\implies x=\cfrac{3\cdot 4.50}{2}[/tex]

For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?

Answers

Didn't you mean y = ax^2? "^" denotes "exponentiation."

The first derivative of y = ax^2 represents the slope of the tangent line to the curve of y = ax^2. 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=ax^2 at (2,4a) is found by (1) differentiating y=ax^2 with respect to x, (2) letting x = 2 in the result: dy/dx = 2ax => dy/dx (at 2,4a) = 2a(2) = 4a

The line 2x + y = b is supposed to be tangent to y = ax^2 at (2,4a).

The slope of 2x + y = b is 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, so if x=-2 and y=-8,

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

Thus, the equation of the parabola is y = ax^2 = (-1/2)x^2.

a = -2 and b = -8 are the required a and b values.

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.

What is the slope of the tangent line?

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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Allison drive 30 mph through the city and 55 mph on the New Jersey Turnpike she drove 90 miles from battery Park to the Jersey shore how much of the time was city driving if she needs about 1.2 hours on the turnpike

Answers

Attached a solution and showed work.

6578 divided 34

20\5

Answers

0.0037942 is your answer

Answer:

263.56 :)

Step-by-step explanation:

Amanda can jog 18 miles in 5 hours. At this rate, how many miles can Amanda jog in 4 hours?

Answers

18 = x
__ __
5 4
x = 18•4 / 5
72/5 = 14.4 final answer

we know that

Amanda can jog [tex]18[/tex] miles in [tex]5[/tex] hours

by proportion

Find the number of miles that Amanda can jog in [tex]4[/tex] hours

[tex]\frac{18}{5} \frac{miles}{hours} =\frac{x}{4} \frac{miles}{hours} \\ \\5*x=18*4 \\ \\x=72/5 \\ \\ x=14.4\ miles[/tex]

therefore

the answer is

[tex]14.4\ miles[/tex]

The measure of arc XZ is 28° what is the measure of XYZ

Answers

Arc XZ is marked in red (see the image attachment)
Angle XYZ is marked in blue

It turns out that angle XYZ is exactly half the measure of arc XZ. This is due to the inscribed angle theorem. Angle XYZ is an inscribed angle that cuts off, or subtends, arc XZ.

measure of arc XZ = 28 degrees
measure of angle XYZ = (1/2)*(measure of arc XZ)
measure of angle XYZ = (1/2)*(28 degrees)
measure of angle XYZ = 14 degrees

Answer: D) 14 degrees

solve 6[4x(72-63)divided3]

Answers

parentheses first, 72-63 is 9 next multiply by 4 which is 36 the you’ll divide by 3 now you have 12 so multiply that by 6 and you get 72. Just a hint, another way to write that without using the multiply or the divided would be 6[4(72-63)/3] but either way you want is fine. Hope this helped :)

The slope of a line is 1, and the y-intercept is -1. What is the equation of the line written in slope-intercept form?

y = x - 1
y = 1 - x
y = -x - 1

Answers

The slope-intercept form is:
y = mx + b
where m = slope, and b = y-intercept.
You need a slope of 1, so m = 1.
You need a y-intercept of -1, so b = -1.
Replace m with 1 and b with -1 in the slope intercept form to get

y = 1x + (-1)

which simplifies to

y = x - 1

Find the second degree Taylor polynomial for f(x)= sqrt(x^2+8) at the number x=1

Answers

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

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Functions

Function Notation

Calculus

Differentiation

DerivativesDerivative Notation

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:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

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

Taylor Polynomials

Approximating Transcendental and Elementary Functions[tex]\displaystyle P_n(x) = \frac{f(c)}{0!} + \frac{f'(c)}{1!}(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \frac{f'''(c)}{3!}(x - c)^3 + ... + \frac{f^n(c)}{n!}(x - c)^n[/tex]

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

[Function] 1st Derivative: [tex]\displaystyle f'(x) = \frac{x}{\sqrt{x^2 + 8}}[/tex][Function] 2nd Derivative: [tex]\displaystyle f''(x) = \frac{8}{(x^2 + 8)^\bigg{\frac{3}{2}}}[/tex]

Step 3: Evaluate

Substitute in center x [Function]: [tex]\displaystyle f(1) = \sqrt{1^2 + 8}[/tex]Simplify: [tex]\displaystyle f(1) = 3[/tex]Substitute in center x [1st Derivative]: [tex]\displaystyle f'(1) = \frac{1}{\sqrt{1^2 + 8}}[/tex]Simplify: [tex]\displaystyle f'(1) = \frac{1}{3}[/tex]Substitute in center x [2nd Derivative]: [tex]\displaystyle f''(1) = \frac{8}{(1^2 + 8)^\bigg{\frac{3}{2}}}[/tex]Simplify: [tex]\displaystyle f''(1) = \frac{8}{27}[/tex]

Step 4: Write Taylor Polynomial

Substitute in derivative function values [Taylor Polynomial]: [tex]\displaystyle P_2(x) = \frac{3}{0!} + \frac{\frac{1}{3}}{1!}(x - c) + \frac{\frac{8}{27}}{2!}(x - c)^2[/tex]Simplify: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - c) + \frac{4}{27}(x - c)^2[/tex]Substitute in center c: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]

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].

Suppose e(x) = 3 and e[x(x − 1)] = 27.5. (a) what is e(x2)? [hint: first verify that e[x(x − 1)] = e[x2 − x] = e(x2) − e(x).]

Answers

52(ex) would be your answer

a neighborhood garden that is 2/3 of an acre is to be divided 4 equal-size sections

Answers

(2/3) / 4 =
2/3 * 1/4 =
2/12 =
1/6.........1/6 of an acre

a fan has 5 equally spaced blades. what is the least number of degrees that can rotate the fan onto self?

Answers

If I understand your question correctly, then I believe your answer is 72 degrees, because 360 divided by 5 equals 72.

72 degree's
Hope this helps!

WILL GIVE BRAINEST
categorize the graph as liner increase...

Answers

Its a linear decrease.

liner decreases is the answer

Solve for the distance between (522, 1322) and (9000, -1337) to the third decimal.

Answers

[tex]\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 522}}\quad ,&{{ 1322}})\quad % (c,d) &({{ 9000}}\quad ,&{{ -1337}}) \end{array} \\\\\\ % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}\\\\ -------------------------------\\\\ d=\sqrt{(9000-522)^2+(-1337-1322)^2}\\\\\\ d\sqrt{8478^2+(-2659)^2} \\\\\\ d=\sqrt{71876484+7070281}\implies d=\sqrt{78946765} \\\\\\ d\approx 8885.199209922\implies d\approx 8885.199[/tex]

6-1. let x have a poisson distribution with a mean of 4. find (a) p(2≤x≤5). (b) p(x≥3). (c) p(x≤3).

Answers

The mean of a Poisson distribution coincides with its rate parameter, so [tex]\lambda=4[/tex]. We have a PMF of

[tex]f_X(x)=\dfrac{e^{-4}4^x}{x!}[/tex]

So,

a. [tex]\mathbb P(2\le X\le5)=\dfrac{e^{-4}4^2}{2!}+\cdots+\dfrac{e^{-4}4^5}{5!}\approx0.6936[/tex]

b. [tex]\mathbb P(X\ge3)=\dfrac{e^{-4}4^3}{3!}+\dfrac{e^{-4}4^4}{4!}+\cdots\approx0.7619[/tex]

c. [tex]\mathbb P(X\le3)=\dfrac{e^{-4}4^0}{0!}+\cdots+\dfrac{e^{-4}4^3}{3!}\approx0.4335[/tex]

Find the missing number of each unit rate

Answers

4/2 = 2/1
6/2 = 3/1
12/4 = 3/1
if the last on is negative it would -24/8 = -3/1 if positive 3/1 hope this helps remember you would reduce your fraction

4/2 reduces to 2/1.

12/4 reduces to 3/1

Can you do the remaining problems?

whats the nearest tenth to -61 square root

Answers

well, the square root of a negaitve number will be an imaginary number
that is, i=√-1

so we can write
[tex]\sqrt{-61}=(\sqrt{-1})(\sqrt{61})=(i)(\sqrt{61})[/tex] [tex] \sim (i)(7.81) \sim 7.8i[/tex]

Hello!
the √(-61) is a complex number √61i

But if you wish to find the value of an irrational and imaginary number, I will give you the approximate answer.

[tex] \sqrt{-61} \to \sqrt{-1*61} \to \sqrt{-1} * \sqrt{61} \to i* \sqrt{61} \to i*7,81024...\: \boxed{\approx 7.8i}[/tex]

Find the volume of a rectangular block of ice 3 feet by 6 1/3 and 1 1/2 feet

Answers

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.

What is the volume of a paperback book 21 cm tall, 12 cm wide, and 3.5 cm thick?

Answers

A book can be considered a rectangular prism. In order to find the volume of a rectangular prism, the width must be multiplied by the height and length. As a result we multiply 21cm x 12cm x 3.5 cm to get a volume of 882cm^2.

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

What is volume?

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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Marco comma roberto comma dominique comma and claricemarco, roberto, dominique, and clarice work for a publishing company. the company wants to send two employees to a statistics conference. to be fair, the company decides that the two individuals who get to attend will have their names drawn from a hat. this is like obtaining a simple random sample of size 2. (a) determine the sample space of the experiment. that is, list all possible simple random samples of size n equals 2n=2. (b) what is the probability that marco and robertomarco and roberto attend the conference? (c) what is the probability that dominiquedominique attends the conferenceattends the conference?

Answers

This seems a bit hard..

Answer:

Yes

Step-by-step explanation:

ye mom ye mom lolololol

the sum of two numbers is 18 . the difference of the two numbers is -2 find the numbers

Answers

a and b is a numbers
a + b = 18
a - b = -2

2a = 16
a = 8

8 + b = 18
b = 10

You make a large bowl of salad to share with your friends. Your brother eats 1/3 of it before they come over. What fractional portion of the original bowl of salad does each friend receive?

Answers

How many friends are there?

Answer:

[tex]\frac{2}{3x}[/tex]

Where, x = total friends

Step-by-step explanation:

Given,

The part of the salad has eaten = [tex]\frac{1}{3}[/tex]

Total part of the salad = 1,

Thus, the remaining part of the salad = original part - part has eaten

[tex]=1-\frac{1}{3}[/tex]

[tex]=\frac{3-1}{3}[/tex]

[tex]=\frac{2}{3}[/tex]

If there are x friends,

Then the portion of the original bowl of salad received by each friend

[tex]=\frac{\text{Remaining part}}{\text{Total friends}}[/tex] [tex]=\frac{2}{3x}[/tex]

determine the quadratic function of f whose graph is given. The vertex is (1,-3) and the y-intercept is -2

Answers

hmmm the y-intercept is at -2? what does that mean? well, is where the graph "intercepts" or touches the y-axis, and when that happens, x = 0, so the point is really ( 0 , -2 ).

and we know where the vertex is at. Let's assume a vertical parabola, in which case the squared variable is the "x".

[tex]\bf \qquad \textit{parabola vertex form}\\\\ \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ vertex \begin{cases} h=1\\ k=-3 \end{cases}\implies y=a(x-1)^2-3 \\\\\\ \textit{now, we also know that } \begin{cases} x=0\\ y=-2 \end{cases}\implies -2=a(0-1)^2-3 \\\\\\ 1=a(-1)^2\implies \boxed{1=a}\qquad thus\implies \begin{cases} y=1(x-1)^2-3\\ \textit{or just}\\ y=(x-1)^2-3 \end{cases} [/tex]

You've got 60 homework problems to do and it took you 10 minutes to do eight of them. At that rate, how long will it take?

Answers

10 minutes : 8 problems
x minutes : 60 problems
10 * 60/8 = 150/2 = 75 minutes for 60 problems

about 75 minutes because if you divide 60 by 8, you get 7.5, 7.5x10=75 so that is your answer

Determine whether the given function is linear. if the function is linear, express the function in the form f(x)

Answers

what is the equation or graph we are supposed to look at

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

MARIA WALKED 4.035 KILOMETERES. WHAT IS 4.035 WRITTEN IN AS EXPANDED FORM ?

Answers

Here's my interpretation: 4 whole km, plus zero tenths of a km, plus 3 hundredths of a km, plus 5 thousandths of a km.

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]

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