Use substitution to solve each system of equations. y = 4x + 22 4x – 6y = –32

(–5, 2)

(2, –5)

(–8, 1)

(4, 7)

Answer:

If you are looking for the dimensions of the playground, they are that the width is 4 yards and the length is 16 yards

Step-by-step explanation:

We need to know 2 things here:  first, the area of a rectangle which is A = l×w,

and then we need to know how to express one dimension in terms of the other, since we have way too many unknowns right now to solve for anything!

We are told that the length is 4 times the width, so if the width is "w", then the length is "4w".  We know the area is 64, so let's sub in those values where they belong in the area formula:

64 = 4w(w).  Multiplying to simplify we get

[tex]64=4w^2[/tex]

The easiest way to do this is to divide both sides by 4 to get

[tex]16=w^2[/tex]

and when you take the square root of 16 you get 4 and -4.  However, the two things in math that will never ever be negative are distance measurements and time.  So the -4 won't do.  That means that w = 4.  If that be the case, and the length is 4 times the width, then the length is 16.  And there you go!

Answer:

  64

Step-by-step explanation:

The value of a+b+c is the value of the expression when x=1:

  (3+5)^2 = 8^2 = 64

Answer:

Minimum Unit Cost = $14,362

Step-by-step explanation:

The standard form of a quadratic is given by:

ax^2 + bx + c

So for our function, we can say,

a = 0.6

b = -108

c = 19,222

We can find the vertex (x-coordinate where minimum value occurs) by the formula -b/2a

So,

-(-108)/2(0.6) = 108/1.2 = 90

Plugging this value into original function would give us the minimum (unit cost):

[tex]c(x)=0.6x^2-108x+19,222\\c(90)=0.6(90)^2-108(90)+19,222\\=14,362[/tex]

Answer:

The minimum unit cost is 14,362

Step-by-step explanation:

The minimum unit cost is given by a quadratic equation. Therefore the minimum value is at its vertex

For a quadratic function of the form

[tex]ax ^ 2 + bx + c[/tex]

the x coordinate of the vertex is

[tex]x=-\frac{b}{2a}[/tex]

In this case the equation is: [tex]c(x) = 0.6x^2-108x+19,222[/tex]

Then

[tex]a= 0.6\\b=-108\\c=19,222[/tex]

Therefore the x coordinate of the vertex is:

[tex]x=-\frac{(-108)}{2(0.6)}[/tex]

[tex]x=90[/tex]

Finally the minimum unit cost is:

[tex]c(90)=0.6(90)^2-108(90)+19,222\\\\c(90)=14,362[/tex]

Answer:

The correct answer is that a possible value for n could be all numbers from 101 to 120.

Step-by-step explanation:

Ok, to solve this problem:

You have that: [tex]10 <\sqrt{n} <11[/tex]

Then, applying the properties of inequations, the power is raised by 2 on both sides of the inequation:

[tex](10)^{2} <(\sqrt{n} )^{2} <(11)^{2}[/tex]

[tex]100<n<121[/tex]

Then, a possible value for n could be all numbers from 101 to 120.

The equation of the tangent to the curve at the point corresponding to the given values of the parametric equations given is;

y - 1 = ¹/₁₀(x - 1)

We are given;

x = cos θ + sin(10θ)

y = sin θ + cos(10θ)

Since we want to find equation of tangent, let us first differentiate with respect to θ. Thus;

dx/dθ = -sin θ + 10cos (10θ)

Similarly;

dy/dθ = cos θ - 10sin(10θ)

To get the tangent dy/dx, we will divide dy/dθ by dx/dθ to get;

(dy/dθ)/(dx/dθ) = dy/dx =  (cos θ - 10sin(10θ))/(-sin θ + 10cos(10θ))

To get the tangent, we will put the angle to be equal to zero.

Thus, at θ = 0, we have;

dy/dx = (cos 0 - 10sin 0)/(-sin 0 + 10cos 0)

dy/dx = 1/10

Also, at θ = 0, we can get the x-value and y-value of the parametric functions.

Thus;

x = cos 0 + sin 0

x = 1 + 0

x = 1

y = sin 0 + cos 0

y = 0 + 1

y = 1

Thus, the equation of the tangent line to the curve in point slope form gives us;

y - 1 = ¹/₁₀(x - 1)

Read more at; https://brainly.com/question/13388803

Final answer:

To find the tangent line to the curve defined by the parametric equations at θ = 0, we compute the derivatives of both x and y with respect to θ, leading to a slope of 1/10. By evaluating the original parametric equations at θ = 0, we find that the tangent passes through (1, 1), resulting in the equation y - 1 = 1/10(x - 1).

Explanation:

To find an equation of the tangent to the given parametric curve at θ = 0, we first need the parametric equations given by x = cos(θ) + sin(10θ) and y = sin(θ) + cos(10θ). To find the slope of the tangent, we compute the derivatives δy/δx = (δy/δθ)/(δx/δθ) at θ = 0.

Computing the derivatives, δx/δθ = -sin(θ) + 10cos(10θ) and δy/δθ = cos(θ) - 10sin(10θ), and plugging in θ = 0, we get δx/δθ = 10 and δy/δθ = 1. Hence, the slope is 1/10. Evaluating the functions at θ = 0 gives x = 1 and y = 1. Thus, the tangent line equation at θ = 0 is y - y_0 = m(x - x_0), which simplifies to y - 1 = 1/10(x - 1).

Answer:

1.6 hours

Step-by-step explanation:

I started off with T(t)=70+Ce^kt

then since the initial temp was 98.6 I did T(0)=98.6=70+C so C=28.6

Then T(1) = 80 = 28.6e^k + 70

k = ln (10/28.6)

Then plugged that into

T(t)=85=28.6e^ln(10/28.6)t + 70

and got t=.61

The answer says it is about 1.6 hours.

The time that has elapsed before the body was found is 1.5 hour

The given parameters;

Apply the Newton's method of cooling equation;

[tex]T(t) = T_{s} + (T_{o} - T_{s})e^{kt}\\\\T(t) = 70 + (98.6 - 70)e^{kt}\\\\T(t) = 70 + 28.6e^{kt}[/tex]

At the time of discovery, we have the following equation,

[tex]T_{t} = 70 + 28.6e^{kt}\\\\80 = 70 + 28.6e^{kt}\\\\10 = 28.6k^{kt}[/tex]

1 hour later, t + 1, we have the second equation;

[tex]75 = 70 + 28.6e^{kt} \\\\5 = 28.6e^{k(t+ 1)} \\\\5 = 28.6e^{kt + k} ---- (2)[/tex]

divide equation 1  by equation 2;

[tex]\frac{10}{5} = \frac{28.6e^{kt}}{28.6 e^{kt + k}} \\\\2 = e^{kt - kt - k}\\\\2 = e^{-k}\\\\-k = ln(2)\\\\k = -0.693[/tex]

The time when he dead body was discovered is calculated as;

[tex]10 = 28.6e^{kt}\\\\10= 28.6e^{-0.693t}\\\\e^{-0.693t} = \frac{10}{28.6} \\\\-0.693 t = ln(\frac{10}{28.6} )\\\\-0.693t = -1.05\\\\t = \frac{1.05}{0.693} \\\\t = 1.515 \ \\\\t \approx 1.5 \ hr[/tex]

Thus, the time that has elapsed before the body was found is 1.5 hour

Learn more here: https://brainly.com/question/15824468

Answer:

x = 1.5

Step-by-step explanation:

The left side of the equation is graphed as a straight line with a slope of -5/3 and a y-intercept of +7. The right side of the equation is graphed as a horizontal line at y = 4.5. The point of intersection of these lines has the x-coordinate of the solution: x = 1.5.

Answer:

2 outcomes

Step-by-step explanation:

Let's list count all the possible outcomes:

(1,3) (1,4) (1,5) (1,6)

(2,3) (2,4) (2,5) (2,6)

(3,3) (3,4) (3,5) (3,6)

As expected, there are 12 (3x4) possible outcomes.

How many outcomes do not show an even number (so showing 1 or 3) on the first spinner and show a 6 on the second spinner?

There are two cases where 6 is on the second spinner and NOT an even number on the first spinner: (1,6) and (3,6)

ANSWER

Zero(s)

[tex]x = 0[/tex]

The function is discontinuous at

[tex]x = - 3 \:and \: x = 3[/tex]

EXPLANATION

The given rational function is

[tex] y = \frac{3x}{ {x}^{2} - 9 } [/tex]

For this function to be equal to zero, then the numerator must be zero.

Equate the numerator to zero and solve for x.

[tex]3x = 0[/tex]

This implies that

[tex]x = \frac{0}{3} = 0[/tex]

The rational function is discontinuous when the denominator is equal to zero.

[tex] {x}^{2} - 9 = 0[/tex]

Solve this quadratic equation using the square root method or otherwise.

[tex] {x}^{2} = \pm \sqrt{9} [/tex]

[tex]{x} = \pm 3[/tex]

There is discontinuity at

[tex]x = - 3 \:and \: x = 3[/tex]

Check the picture below.

[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ P(\stackrel{x_1}{-10}~,~\stackrel{y_1}{-2})\qquad Q(\stackrel{x_2}{4}~,~\stackrel{y_2}{6}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{4-10}{2}~~,~~\cfrac{6-2}{2} \right)\implies \left( \cfrac{-6}{2}~,~\cfrac{4}{2} \right)\implies \stackrel{\textit{center}}{(-3~,~2)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \stackrel{\textit{center}}{(\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})}\qquad Q(\stackrel{x_2}{4}~,~\stackrel{y_2}{6})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{radius}{r}=\sqrt{[4-(-3)]^2+[6-2]^2}\implies r=\sqrt{(4+3)^2+(6-2)^2} \\\\\\ r=\sqrt{49+16}\implies r=\sqrt{65} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{-3}{ h},\stackrel{2}{ k})\qquad \qquad radius=\stackrel{\sqrt{65}}{ r} \\[2em] [x-(-3)]^2+[y-2]^2=(\sqrt{65})^2\implies (x+3)^2+(y-2)^2=65[/tex]

Part A

Yes, triangle ABC and triangle APQ are similar because of Angle-Angle similarity.

Angle BAC is congruent to Angle PAQ because of reflexive property (they share the same angle).

It is given that Segment BC is parallel to Segment PQ, so Angle ABC is congruent to Angle APQ because the corresponding angles postulate.

Part B

Segment PQ corresponds to Segment BC because they are parallel to each other.

Part C

Angle APQ corresponds to Angle B because of the corresponding angles postulate.

ANSWER

B. 3 ft by 8 ft

EXPLANATION

The area is given as 24 square feet.

This implies that,

[tex]l \times w = 24[/tex]

The perimeter of the rectangular field is given as 22 feet.

This implies that,

[tex]2(l + w) = 22[/tex]

Or

[tex]l + w = 11[/tex]

We make w the subject in this last equation and put it inside the first equation.

[tex]w = 11 - l[/tex]

When we substitute into the first equation we get;

[tex]l(11 - l) = 24[/tex]

[tex]11l - {l}^{2} = 24[/tex]

This implies that,

[tex] {l}^{2} - 11l + 24 = 0[/tex]

[tex](l - 3)(l - 8) = 24[/tex]

[tex]l = 3 \: or \: 8[/tex]

When l=3, w=24

Therefore the dimension is 3 ft by 8 ft

Answer:

The correct answer is option B.  3 ft  by 8 ft

Step-by-step explanation:

Points to remember

Area of rectangle = length * breadth

Perimeter of rectangle = 2(Length + Breadth)

It is given that, The area of a rectangular flower bed is 24 square feet. The perimeter of the same flower bed is 22 feet

To find the correct option

1). Check option A

Area = 2 * 12 = 24

Perimeter = 2( 2 + 12 ) = 28

False

2) Check option B

Area = 3 * 8 = 24

Perimeter = 2(3  + 8 ) = 22

True

3). Check option C

Area = 3 * 6 = 18

Perimeter = 2( 3 + 6 ) = 18

False

4). Check option D

Area = 4 * 6 = 24

Perimeter = 2( 4 +6 ) = 20

False

The correct answer is option B.  3 ft  by 8 ft

Step-by-step explanation:

Remember that in a linear function of the form [tex]f(x)=mx+b[/tex], [tex]m[/tex] is the slope and [tex]b[/tex] is the why intercept.

Part A. Since [tex]g(x)=2x+6[/tex], its slope is 2 and its y-intercept is 6

Now, to find the slope of [tex]f(x)[/tex] we are using the slope formula:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

where

[tex]m[/tex] is the slope

[tex](x_1,y_1)[/tex] are the coordinates of the first point

[tex](x_2,y_2)[/tex] are the coordinates of the second point

From the table the first point is (-1, -12) and the second point is (0, -6)

Replacing values:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{-6--(12)}{0-(-1)}[/tex]

[tex]m=\frac{-6+12}{0+1}[/tex]

[tex]m=6[/tex]

The slope of f(x) is bigger than the slope of g(x), which means the line represented by f(x) is stepper than the line represented by g(x).

Part B. To find the y-intercept of f(x) we are taking advantage of the fact that the y-intercept of a linear function occurs when x = 0, so we just need to look in the table for the value of f(x) when x = 0. From the table [tex]f(x)=-6[/tex] when [tex]x=0[/tex]; therefore the y-intercept of [tex]f(x)[/tex] is -6.

We already know that the y-intercept of g(x) is 2. Since 2 is bigger than -6, function g(x) has a greater y-intercept.

Answer:

g(x) = x^2 - 11x + 26

Step-by-step explanation:

In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.

Given the function;

f(x) = x2 - 3x - 2

a shift 4 units to the right implies that we shall be subtracting the constant 4 from the x values of the function;

g(x) = f(x-4)

g(x) = (x - 4)^2 - 3(x - 4) -2

g(x) = x^2 - 8x + 16 - 3x + 12 - 2

g(x) = x^2 - 11x + 26

Answer: Option A

Step-by-step explanation:

Given the parent function [tex]f(x)=x^5[/tex], it can be transformated:

If  [tex]f(x)=x^5-k[/tex], then the function is shifted k units down.

If  [tex]f(x)=a(x^5)[/tex] and [tex]a > 1[/tex]  it is vertically stretched it, but if [tex]0 < a < 1[/tex] it is vertically compressesd.

If  [tex]f(x)=-(x^5)[/tex], then the function is reflected over the x-axis.

Then, if the function given is reflected over the x-axis, it is vertically streteched by a factor o 2 and it is shifted down 1 units, the function that results after this transformations is:

[tex]g(x)=-2(x^5)-1[/tex]

[tex]g(x)=-2x^5-1[/tex]

In the [tex]x[/tex]-[tex]y[/tex] plane, the base has equation(s)

[tex]16x^2+9y^2=144\implies y=\pm\dfrac43\sqrt{9-x^2}[/tex]

which is to say, the distance (parallel to the [tex]y[/tex]-axis) between the top and the bottom of the ellipse is

[tex]\dfrac43\sqrt{9-x^2}-\left(-\dfrac43\sqrt{9-x^2}\right)=\dfrac83\sqrt{9-x^2}[/tex]

so that at any given [tex]x[/tex], the cross-section has a hypotenuse whose length is [tex]\dfrac83\sqrt{9-x^2}[/tex].

The cross-section is an isosceles right triangle, which means the legs occur with the hypotenuse in a ratio of 1 to [tex]\sqrt2[/tex], so that the legs have length [tex]\dfrac8{3\sqrt2}\sqrt{9-x^2}[/tex]. Then the area of each cross-section is

[tex]\dfrac12\left(\dfrac8{3\sqrt2}\sqrt{9-x^2}\right)\left(\dfrac8{3\sqrt2}\sqrt{9-x^2}\right)=\dfrac{16}9(9-x^2)[/tex]

Then the volume of this solid is

[tex]\displaystyle\frac{16}9\int_{-3}^39-x^2\,\mathrm dx=\boxed{64}[/tex]

Solid [tex]\( S \)[/tex] has elliptical base[tex]\( 16x^2 + 9y^2 = 144 \)[/tex]. Triangular cross-sections yield volume [tex]\( 128 \)[/tex] cubic units.

let's break it down step by step.

1. Understanding the Solid: The solid [tex]\( S \)[/tex] has a base in the shape of an ellipse given by the equation [tex]\( 16x^2 + 9y^2 = 144 \).[/tex] The cross-sections perpendicular to the x-axis are isosceles right triangles with their hypotenuse lying on the base ellipse.

2. **Equation of the Ellipse**: To understand the shape of the base, let's rearrange the equation of the ellipse to find [tex]\( y \)[/tex]  in terms of [tex]\( x \):[/tex]

 [tex]\[ 16x^2 + 9y^2 = 144 \] \[ y^2 = \frac{144 - 16x^2}{9} \] \[ y = \pm \frac{4}{3} \sqrt{9 - x^2} \][/tex]

3. Finding the Length of the Hypotenuse: The length of the hypotenuse of each triangle is twice the value of [tex]\( y \)[/tex] at any given point on the ellipse. So, the length [tex]\( h \)[/tex] of the hypotenuse is given by:

  [tex]\[ h = \frac{8}{3} \sqrt{9 - x^2} \][/tex]

4. Area of Each Cross-Section Triangle: The area of each cross-section triangle is [tex]\( \frac{1}{2} \times \text{base} \times \text{height} \),[/tex] where the base is the same as the height. So, the area is:

[tex]\[ \text{Area} = \frac{1}{2} \times \frac{8}{3} \sqrt{9 - x^2} \times \frac{8}{3} \sqrt{9 - x^2} = \frac{32}{9} (9 - x^2) \][/tex]

5. Integrating to Find Volume: To find the volume of the solid, we integrate the area function over the interval that covers the base ellipse, which is [tex]\([-3, 3]\)[/tex] in this case.

  [tex]\[ V = \int_{-3}^{3} \frac{32}{9} (9 - x^2) \, dx \][/tex]

6. Solving the Integral: Integrating [tex]\( (9 - x^2) \)[/tex] with respect to[tex]\( x \)[/tex]  yields:

  [tex]\[ = \frac{32}{9} \int_{-3}^{3} (9 - x^2) \, dx \] \[ = \frac{32}{9} \left[ 9x - \frac{x^3}{3} \right]_{-3}^{3} \] \[ = \frac{32}{9} \left[ (27 - 9) - (-27 + 9) \right] \] \[ = \frac{32}{9} \times 36 \] \[ = \frac{1152}{9} \] \[ = 128 \][/tex]

7. Final Result: So, the volume of the solid [tex]\( S \)[/tex] is [tex]\( 128 \)[/tex] cubic units.

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is .54

Probability is the likeliness of the occurrence of an event.

Let :

P(A) = Probability of feeling guilty about wasting food = .39

P(B) = Probability of feeling guilty about leaving lights on = .27

P(A∩B) = Probability of feeling guilty for both of these reasons = .12

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is :

P(A∪B) = P(A) + P(B) - P(A∩B)

P(A∪B) = .39 + .27 - .12

Different Birthdays: https://brainly.com/question/7567074

Dependent or Independent Events: https://brainly.com/question/12029535

Grade: High School

Subject: Mathematics

Chapter: Probability

Keywords: Person, Probability, Outcomes, Random, Event, Room, Wasting, Food

Explanation:

Multiply it out.

n^2 -n -(n^2 +5n+6)

= -6n -6

= -6(n +1)

For any integer value of n, this is divisible by 6. (The quotient is -(n+1).)

Answer:

Part 1) The larger integer is 11

Part 2) The denominator is 5

Part 3) The positive integer is 4

The graph in the attached figure

Step-by-step explanation:

Part 1)

Let

x----> the smaller positive integer

y-----> the larger positive integer

we know that

[tex]x^{2} +y^{2} =185[/tex] -----> equation A

[tex]x=y-3[/tex] -----> equation B

substitute equation B in equation A and solve for y

[tex](y-3)^{2} +y^{2} =185\\ \\y^{2} -6y+9+y^{2}=185\\ \\2y^{2}-6y-176=0[/tex]

using a graphing calculator-----> solve the quadratic equation

The solution is y=11

[tex]x=11-3=8[/tex]

Part 2)

Let

x----> the numerator of the fraction

y-----> the denominator of the fraction

we know that

[tex]x=2y+1[/tex] ----> equation A

[tex]\frac{x+4}{y+4}=\frac{5}{3}[/tex] ----> equation B

substitute equation A in equation B and solve for y

[tex]\frac{2y+1+4}{y+4}=\frac{5}{3}[/tex]

[tex]\frac{2y+5}{y+4}=\frac{5}{3}\\ \\6y+15=5y+20\\ \\6y-5y=20-15\\ \\y=5[/tex]

[tex]x=2(5)+1=11[/tex]

Part 3)

Let

x----> the positive integer

we know that

[tex]x-\frac{1}{x}=\frac{15}{4}[/tex]

solve for x

[tex]x-\frac{1}{x}=\frac{15}{4}\\ \\4x^{2}-4=15x\\ \\4x^{2}-15x-4=0[/tex]

using a graphing calculator-----> solve the quadratic equation

The solution is x=4

Final answer:

The variables in this problem represent the scores in the score matrix S. The objective for this 0-1 integer linear program is to maximize the overall score. The entire 0-1 integer linear program in standard form includes constraints to ensure that each detection is assigned to a single track and each track is assigned only once, and there are N + M constraints in total.

Explanation:

(a) In this problem, the variables represent the scores in the score matrix S. There are N detections and M tracks, so we have N rows and M columns in the score matrix.

(b) The objective for this 0-1 integer linear program is to maximize the overall score, which is the sum of the selected detections' scores.

(c) The entire 0-1 integer linear program in standard form can be defined as:

There are N + M constraints in total.

Answer:

4.382 cubiti

Step-by-step explanation:

That's a simple exercise of cross-multiplication:

[tex]\frac{x}{1.95}  = \frac{1}{0.445}[/tex]

x being the measure in cubitus we're looking for. We can isolate it:

x = (1.95 m * 1 ) / (0.445 m/cubitus) = 4.382 cubiti

1.95 m = 4.382 cubiti

Which totally makes sense... since a cubitus is roughly half a meter long... and the basketball is 2 meters high... so there are roughly 4 cubiti in 2 meters.

The height of the basketball forward in cubiti is approximately 4.38.

To convert the height from meters to cubiti, we use the conversion factor provided in the question:

1 cubitus = 0.445 m

Given the height of the basketball forward is 1.95 m, we divide this value by the conversion factor to find the height in cubiti:

Height in cubiti = Height in meters / Conversion factor

Height in cubiti = 1.95 m / 0.445 m/cubitus

Now, we perform the division:

Height in cubiti ≈ 4.38 cubiti

Since the value of 1 cubitus is considered to be exact, the number of significant figures in the answer is determined by the height in meters, which is 1.95 m (three significant figures). Therefore, the answer is rounded to three significant figures as well.

Answer:

6t+3

Step-by-step explanation:

If t represents the number of tens, then 6t is six times the number of tens. 3 more than that is ...

6t+3

Answer:

6t + 3

Step-by-step explanation:

Given: Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens

To Find: Let t represent the number of tens. Write an expression for the number of fives.

Solution:

Total number of ten bills are = [tex]\text{t}[/tex]

As given in question,

The number of fives is three more than six times the number of tens

therefore

total number of fives are

                                          =[tex]6\text{t}+3[/tex]

here,  t represents total number of $5 and $10 bills Jeannette has in her wallet

Final expression for total number of [tex]\$5[/tex] bills is [tex]6\text{t}+3[/tex]

Taking the cubic root of a number is the same as raising that number to the power of 1/3.

Moreover, we have

[tex]64 = 2^6[/tex]

So, we have

[tex]\sqrt[3]{64} = \sqrt[3]{2^6} = (2^6)^{\frac{1}{3}} = 2^{6\cdot\frac{1}{3}} = 2^2 = 4 [/tex]

Answer:

4

Step-by-step explanation:

Since we see a cube root, we will attempt to rewrite 64 as a number with an exponent of 3.

[tex]\sqrt[3]{64}[/tex]

[tex]= \sqrt[3]{4^3}[/tex]

[tex]= 4 [/tex]

The answer is 5 hope this helps

Answer:

5

Step-by-step explanation:

abs(a + b) + abs(c)

abs(-3 + 7) + abs(1)

abs(4) + abs(1)

4 + 1

5

Answer:

about 10.06 oz.

Step-by-step explanation:

Let x represent the number of ounces of pure gold you need to add. Then the amount of gold in the mix is ...

100%·x + 75%·5 = 91.7%·(x+5)

8.3%·x = 5·16.7% . . . . . . subtract 91.7%·x +75%·5

x = 5 · 16.7/8.3 . . . . . . . . divide by the coefficient of x

x ≈ 10.06 . . . . oz

_____

Alternate solution

The amount of non-gold in the given material is 25%·5 oz = 1.25 oz. That is allowed to be 8.3% of the final mix, so the weight of the final mix will be ...

(1.25 oz)/0.083 ≈ 15.06 oz

Since that weight will include the 5 oz you already have, the amount of pure gold added must be ...

15.06 oz - 5 oz = 10.06 oz

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Comment on these answers

If you work directly with carats instead of percentages, you find the amount of pure gold you need to add is 10.00 ounces, double the amount you have.

The volume of the solid generated by revolving the plane region about the y-axis is approximately 35,929.77 cubic units.

Here,

To use the shell method to find the volume of the solid generated by revolving the plane region bounded by the curves [tex]y = x^{(5/2)}, y = 32[/tex],

and x = 0 about the y-axis, we need to integrate the circumference of cylindrical shells along the y-axis.

The volume V can be expressed as the integral of the circumference of the cylindrical shells from y = 0 to y = 32:

V = ∫[0 to 32] 2π * x * h(y) dy

where h(y) represents the height (or thickness) of each shell, and x is the distance from the y-axis to the curve [tex]y = x^{(5/2)[/tex].

To find h(y), we need to express x in terms of y by rearranging the equation [tex]y = x^{(5/2)[/tex]:

[tex]x = y^{(2/5)[/tex]

Now, we can express the volume integral:

V = ∫[0 to 32] 2π * [tex]y^{(2/5)[/tex] * (32 - y) dy

Now, we'll evaluate the integral:

V = 2π ∫[0 to 32] ([tex]32y^{(2/5)} - y^{(7/5)[/tex]) dy

Integrate each term separately:

[tex]V = 2\pi [(32 * (5/7) * y^{(7/5)}) - (5/12) * y^{(12/5)}] | [0 to 32]\\V = 2\pi [(32 * (5/7) * (32)^{(7/5)}) - (5/12) * (32)^{(12/5)}] - [0][/tex]

Now, evaluate the expression:

[tex]V = 2\pi [(32 * (5/7) * 2^7) - (5/12) * 2^{12}][/tex]

V = 2π [(32 * 1280/7) - (5/12) * 4096]

V = 2π [81920/7 - 341.33]

V ≈ 2π * 81920/7 - 2π * 341.33

V ≈ 36608π - 678.13

The volume of the solid generated by revolving the plane region about the y-axis is approximately 35,929.77 cubic units.

To know more about integral:

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Final answer:

The shell method is used to evaluate the volume of a solid created by revolving a region around the y-axis using a vertical shell element, integrating from x=0 to the x value corresponding to y=32.

Explanation:

To use the shell method to find the volume of the solid generated by revolving the given plane region about the y-axis, we consider a vertical element or 'shell' at a certain x-value with thickness dx. Given the equations [tex]x^{5/2}[/tex], y = 32, and x = 0, these will be the bounds for our region.

The volume of each infinitesimal shell with radius x and height [tex](32 - x^{5/2})[/tex], when revolved around the y-axis, is [tex]2πx(32 -x^{5/2})dx[/tex]. To find the total volume, we integrate this expression from x=0 to where y is 32, which corresponds to the x value where [tex]x^{5/2} = 32[/tex].

Using the substitution [tex]x^{5/2}[/tex] to solve for dx, we get the integral in terms of y, which simplifies the computation. Finally, we evaluate the definite integral to find the volume of the solid of revolution.

The answer is:

The difference between the circle and the square is:

[tex]Difference=4\pi -8[/tex]

To solve the problem, we need to find the area of the circle and the area of the square, and then, subtract them.

For the square we have:

[tex]side=2\sqrt{2}[/tex]

We can calculate the diagonal of a square using the following formula:

[tex]diagonal=side*\sqrt{2}[/tex]

So,

[tex]diagonal=2\sqrt{2}*\sqrt{2}=2*(\sqrt{2})^{2}=2*2=4units[/tex]

The area will be:

[tex]Area_{square}=side^{2}= (2\sqrt{2})^{2} =4*2=8units^{2}[/tex]

For the circle we have:

[tex]radius=\frac{4units}{2}=2units[/tex]

The area will be:

[tex]Area_{Circle}=\pi *radius^{2}=\pi *2^{2}=\pi *4=4\pi units^{2}[/tex]

[tex]Area_{Circle}=4\pi units^{2}[/tex]

Then, the difference will be:

[tex]Difference=Area_{Circle}-Area{Square}=4\pi -8[/tex]

Have a nice day!

ANSWER

[tex]4\pi - 8[/tex]

EXPLANATION

The diagonal of the square can be found

using Pythagoras Theorem.

[tex] {d}^{2} = {(2 \sqrt{2} )}^{2} + {(2 \sqrt{2} )}^{2} [/tex]

[tex]{d}^{2} = 4 \times 2+ 4 \times 2[/tex]

[tex]{d}^{2} = 8+ 8[/tex]

[tex]{d}^{2} = 16[/tex]

Take positive square root

[tex]d = \sqrt{16} = 4[/tex]

The radius is half the diagonal because the diagonal formed the diameter of the circle.

Hence r=2 units.

Area of circle is

[tex]\pi {r}^{2} =\pi \times {2}^{2} = 4\pi[/tex]

The area of the square is

[tex] {l}^{2} = {(2 \sqrt{2)} }^{2} = 4 \times 2 = 8[/tex]

The difference in area is

[tex]4\pi - 8[/tex]

Answer:

2947 lb

Step-by-step explanation:

Find the volume of the sphere

v=4/3 ×pi×r³

r=2ft and pi=3.14

v=4/3 × 3.14×2³

v=33.49 ft³

Given that;

Density ⇒ 88 lb/ft³

Volume⇒33.49 ft³

Mass=?-------------------------------------find the mass

But we know density=mass/volume -----so mass=density × volume

Mass= 88×33.49 =2947.41 pounds

                           ⇒2947 lb

Answer:

A shows an original investment of $200

Step-by-step explanation:

If you plug in x=0, you will get the value of the original investment

When you plug x=0 into A  you get

[tex]y=200(1.02)^{0}[/tex]

This simplifies to

[tex]y=200(1)[/tex]

And finally to

[tex]y=200[/tex]

Answer: First option.

Step-by-step explanation:

You know that the meaning of the symbol of the inequality "<" is: Less than.

So, you can check each option to find the number that could be a length of this piece of wood.

Given [tex]w<5[/tex], you can substitute each number given in the options into this inequality. Then:

[tex]1)\ w<5\\\\4.5<5\ (This\ is\ true)[/tex]

[tex]2)\ w<5\\\\6<5 (This\ is\ not\ true)[/tex]

[tex]3)\ w<5\\\\11.3<5\ (This\ is\ not\ true)[/tex]

[tex]4)\ w<5\\\\13<5\ (This\ is\ not\ true)[/tex]

Therefore, a lenght of the piece of wood could be 4.5