A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 123 inches. Find the dimensions of the package of maximum volume that can be sent. (The cross section is circular.)

The surface area of the composite figure is 444m².

Given a composite figure which is shown in the question.

The  area of ​​a shape (in square units) is the number of unit squares required to cover the entire area without gaps or overlaps. If a hologram has planes, those faces are called faces.

Firstly, we will find the area of front and end triangles by using the formula

Area=(1/2)×b×h and we will multiply this area by 2 because we are finding the area of two same triangles.

Here, h=6 and b=16 and substitute these values in the formula, we get

A₁=2×(1/2)×16×6

A₁=2×8×6

A₁=96m²

Now, we will find the area of the left and right side rectangles which joined both the triangles.

We will find the area by using the formula Area=l×b and we will multiply this area by 2 because we are finding the area of two same rectangles.

here, l=10 and b=5 and substitute these values in the formula, we get

A₂=2×10×5

A₂=100m²

Further, we will find the area of the front and end side rectangles that joined both by the base of the triangles.

We will find the area by using the formula Area=l×b and we will multiply this area by 2 because we are finding the area of two same rectangles.

here, l=16 and b=4 and substitute these values in the formula, we get

A₃=2×16×4

A₃=128m²

Furthermore,  we will find the area of the left and right side rectangles which joined by the front and end rectangles.

We will find the area by using the formula Area=l×b and we will multiply this area by 2 because we are finding the area of two same rectangles.

here, l=4 and b=5 and substitute these values in the formula, we get

A₄=2×4×5

A₄=40m²

Now, we will find the area of the base of the composite figure which is rectangle.

We will find the area by using the formula Area=l×b.

here, l=16 and b=5 and substitute these values in the formula, we get

A₅=16×5

A₅=80m²

So, the surface area of the given composite figure will be

Surface area=A₁+A₂+A₃+A₄+A₅

Surface area=96m²+100m²+128m²+40m²+80m²

Surface area=444m²

Hence, the surface area of the given composite figure is 444m².

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Answer:

One side is 40 cm and the adjacent side is 50 cm

Step-by-step explanation:

We can use a rectangle because a rectangle is a parallelogram.  Draw a rectangle.  Write the width as x and the length as x + 10.  We find the perimeter by adding all 4 side lengths.

Two side lengths are x and two side lengths are x + 10.  If we add all of this together we get 180

x +x +x+10+x+10 = 180

4x + 20 = 180  Combine the like terms.  There are 4 x's and 10+10 is 20

4x = 160  Subtract 20 from both sides

x  = 40  Divide both sides by 4

So, two sides are 40 and two sides are 50, this will add up to 180

Answer:

40 cm & 50 cm

Step-by-step explanation:

one side = X cm (??)

the other side =X+10 cm

Solve

2(x+x+10)=180

x+x+10=90

2x=80

x = 80/2

x=40

therefore

x+10=50cm

Answer:

Step-by-step explanation:

true

The purchase price of the new car if the sales tax is $3500 is $50000

Let the amount of sales tax be A and the purchase price be p

If the amount of sales tax is directly proportional to the purchase price, this is expressed as:

S = kp

If S = $1750 and p = $25000, the;

k = S/p

k = 1750/25000

k = 175/2500

If the sales tax is $3500, the purchase price will be given as:

S = kp
3500 = 175/2500 p

175p = 3500*2500
175p = 8750000

p = 8750000/175

p = 50000

Hence the purchase price of the new car if the sales tax is $3500 is $50000

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The best description of the geometric construction of the quotient z/w on the complex plane is; Option A; z is scaled by a factor of One-fifth and rotated 90 degrees clockwise

We are given;

z = 3(cos(15°) + i sin(15°))

w = 5(cos(90°) + i sin(90°))

Now, if we want to find the quotient z/w, it is clear that in geometric construction, the procedure will be to scale z by a factor of 1/5 and thereafter we will rotate by 90° clockwise.

Thus, option A is the correct answer.

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The conversion factor of the chicken recipe used at the banquet is 4.5.

An equation is an expression that shows the relationship between two or more variables and numbers.

Conversion factor = (new amount * portion size) / (old amount * portion size)

Hence:

Conversion factor = (60 * 3) / (10 * 4) = 4.5

The conversion factor of the chicken recipe used at the banquet is 4.5.

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Answer:

B.) The graph of g(x) is shifted 4 units up

Step-by-step explanation:

In a linear function in slope-intercept form, b, or 4 in this case, represents the y-intercept which technically acts as a vertical shift

Answer: 73

Step-by-step explanation:

I think she's 73? You have to take multiples of nine and see if they work! I did 72 since I knew that 72 goes into 9 equally and if she is 73 there would be a remainder of 1. After that, I stuck with 73 and since I knew that 18 times 4 is also 72, I knew that 73 divided by 4 would give me 18 with a remainder of 1.

The solution is 37 years, that is, Mrs. Laura's age is 37 years old.

Given information in the problem:

1. Mrs. Laura's, L age is between 50 and 80.

2. When you divide age by 9, the remainder is 1.

3. When you divide age by 4, the remainder is also 1.

To find L exact age, start by checking multiples of 9 that leave a remainder of 1 when divided by 4:

9 × 1 + 1 = 10 (remainder of 1 when divided by 4)

9 × 2 + 1 = 19 (remainder of 1 when divided by 4)

9 × 3 + 1 = 28 (remainder of 1 when divided by 4)

9 × 4 + 1 = 37 (remainder of 1 when divided by 4)

9 × 5 + 1 = 46 (remainder of 2 when divided by 4)

9 × 6 + 1 = 55 (remainder of 3 when divided by 4)

9 × 7 + 1 = 64 (remainder of 0 when divided by 4)

Check multiples of 4 that leave a remainder of 1 when divided by 9:

4 × 1 + 1 = 5 (remainder of 1 when divided by 9)

4 × 2 + 1 = 9 (remainder of 1 when divided by 9)

4 × 3 + 1 = 13 (remainder of 4 when divided by 9)

4 × 4 + 1 = 17 (remainder of 8 when divided by 9)

From the above calculations, see that the only common number between the two sets of multiples is 37.

So, L's age is 37 years old.

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A cone is a 3-dimensional shape that consists of a circular or flat surface and a curved side. Thus the volume of the given cone is 4.1 [tex]mi.^{3}[/tex]

A cone is a 3-dimensional shape that consists of a circular or flat surface and curved side. Its volume can be determined by;

The volume of a cone = [tex]\frac{1}{3}[/tex] [tex]\pi[/tex][tex]r^{2}[/tex]h

where r is the radius of its flat surface, and h is its height.

From the given question, we have to determine the value of h using the Pythagoras theorem.

Such that;

[tex]/Hyp/^{2}[/tex] = [tex]/Adj 1/^{2}[/tex] + [tex]/Adj 2/^{2}[/tex]

[tex]4^{2}[/tex] = [tex]h^{2}[/tex] + [tex]1^{2}[/tex]

16 = [tex]h^{2}[/tex] + 1

[tex]h^{2}[/tex] = 15

h = [tex]\sqrt{15}[/tex]

Thus the volume of the cone = [tex]\frac{1}{3}[/tex] [tex]\pi[/tex][tex]r^{2}[/tex]h

                             =  [tex]\frac{1}{3}[/tex]x [tex]\frac{22}{7}[/tex] x [tex]1^{2}[/tex]x    [tex]\sqrt{15}[/tex]

                             = 4.0574

Therefore the volume of the given cone is 4.1 [tex]mi.^{3}[/tex]

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Answer:

The system of equation y=120+20x and y=30x

Step-by-step explanation:

Let the number of days be x and y be the total number minutes

Jamie has walked 120 mins. She will walk at a rate of 20 mins per day.

y=120+20x

Owen walks 30 mins a day. So

y = 30x

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Answer:

8.048 liters

Step-by-step explanation:

Convert 8542 ml to liters:  (8542 ml)(1 liter/1000 ml) = 8.542 liters

Now subtract 0.458 liters to find the amount that a fish tank can hold:

 8.542 liters

-0.458  liters

8.048 liters is the fish tank capacity

F is located on the number 4.5 on the number line.

We know that point D is at -6

Point E is at 8.

Point F is between D and E, such that the ratio:

DF:FE is 3:1

So if we divide the distance between D and E in 4 parts, 3 of these parts are DF, and one of these parts is FE.

First, the distance between E and D is:

distance = 8 - (-6) = 14 units.

Now, if we divide that by 4, we get:

14/4 = 3.5

Then we have:

DF = 3*(3.5) = 10.5

This means that F is at 10.5 units to the right of D, then:

F = D + 10.5 = -6 + 10.5 = 4.5

F is located on the number 4.5 on the number line.

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Answer:

4.5 is the answer :D

Step-by-step explanation:

Answer: 4/7

Step-by-step explanation:

You pick one black sock, meaning there are 14 socks remaining in the drawer, 8 of which are black. Thus, the probability is 8/14, which simplifies to 4/7.

The resulted integral is [tex]I=\frac{8}{3} \times \frac{5 \pi}{16}=\frac{5 \pi}{6}[/tex].

In mathematics, an integral is either a number representing the region under a function's graph for a certain interval or just an added to the initial, the derivative of which is initial function (indefinite integral).

Computation of the integrals:

Step 1: We employ the equations in cylindrical coordinates.

[tex]x=r \cos \theta, y=r \sin \theta, z=z[/tex]

Thus, in cylindrical coordinate system,

E lies above the paraboloid [tex]z=r^{2}[/tex] and below the plane [tex]z=2 r \sin \theta[/tex] .

Therefore, the top part E is  [tex]z=2 r \sin \theta[/tex] is the cross-section between paraboloid and the plane.

Now, at the cross-section use, [tex]r^{2}=2 r \sin \theta[/tex] and  [tex]z=2 r \sin \theta[/tex] .

Thus, the limits are given as ;

[tex]r^{2} \leq z \leq 2 r \sin \theta \quad 0 \leq r \leq 2 \sin \theta[/tex]

Apply the limits as compute the integration;

[tex]\begin{aligned}I=\iiint_{E} z d V &=\int_{0}^{\pi} \int_{0}^{2 \sin \theta} \int_{\tau^{2}}^{2 r \sin \theta} z r d r d z d \theta \\&=\int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[\frac{z^{2}}{2}\right]_{r^{2}}^{2 r \sin \theta} r d r d \theta \\&=\frac{1}{2} \int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[4 r^{2} \sin ^{2} \theta-r^{4}\right] r d r d \theta\end{aligned}[/tex]

                       [tex]\begin{aligned}&=\frac{1}{2} \int_{0}^{\pi} \int_{0}^{2 \sin \theta}\left[4 r^{3} \sin ^{2} \theta-r^{5}\right] d r d \theta \\&=\frac{1}{2} \int_{0}^{\pi}\left[r^{4} \sin ^{2} \theta-\frac{r^{6}}{6}\right]_{0}^{2 \sin \theta} d \theta \\&=\frac{8}{3} \int_{0}^{\pi} \sin ^{6} \theta d \theta\end{aligned}[/tex]

Step 2: Now, calculate for the [tex]I_{1}=\int_{0}^{\pi} \sin ^{6} \theta d \theta[/tex].

 [tex]\begin{aligned}\sin ^{6} \theta &=\left(\sin ^{2} \theta\right)^{2} \times \sin ^{2} \theta \\&=\left[\frac{1-\cos 2 \theta}{2}\right]^{2} \times\left[\frac{1-\cos 2 \theta}{2}\right] \\&=\frac{1}{8}\left(1-2 \cos 2 \theta+\cos ^{2} 2 \theta\right)(1-2 \cos 2 \theta) \\&=\frac{1}{8}\left(1-2 \cos 2 \theta+\frac{1+\cos 4 \theta}{2}\right)(1-2 \cos 2 \theta)\end{aligned}[/tex]

         [tex]\begin{aligned}&=\frac{1}{16}(3-4 \cos 2 \theta+\cos 4 \theta)(1-2 \cos 2 \theta) \\&=\frac{1}{32}(10-15 \cos 2 \theta+6 \cos 4 \theta-\cos 6 \theta)\end{aligned}[/tex]

Further compute the value of

   [tex]\begin{aligned}I_{1} &=\int_{0}^{\pi} \sin ^{6} \theta d \theta \\&=\frac{1}{32} \int_{0}^{\pi}(10-15 \cos 2 \theta+6 \cos 4 \theta-\cos 6 \theta) d \theta \\&=\frac{1}{32}\left[10 \theta-\frac{15 \sin 2 \theta}{2}+\frac{3 \sin 4 \theta}{2}-\frac{\sin 6 \theta}{6}\right]_{0}^{\pi} \\&=\frac{5 \pi}{16}\end{aligned}[/tex]

Therefore, the obtained integral is  [tex]I=\frac{8}{3} \times \frac{5 \pi}{16}=\frac{5 \pi}{6}[/tex].

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The complete question is -

Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate ∫∫∫E z dV, where E lies above the paraboloid

z = x² + y²

and below the plane z = 2y. Use either the Table of Integrals or a computer algebra system to evaluate the integral.

Answer:

it is False

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

The single transformation which takes shape A to shape B as in the task content is a 90° clockwise rotation and a 6 units shift rightward.

By observation, it follows that the orientation of the shape A differs from that of B as it lags by 90° in the clockwise direction.

Additionally, the resulting image from a 90° clockwise rotation about the left vertice of A still reguires that A be shifted 6 units rightwards to yield shape B.

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Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex.  This can be obtained using finding y intercept of both parabolas and vertex of parabola A.

Parabola A, (x + 3)² = y = x² + 6x + 9

Parabola B, y = mx + 9

x=0, y = 9

y coordinate of parabola A is (0,9)

⇒The vertex coordinate,

(-b/2a , -(b²-4ac)/4a) = (-3,0)          

x=0, y = 9  

y coordinate of parabola B is (0,9)    

Isabel claims one solution to the system of two equations must always be the vertex of parabola A which is wrong.  

Hence Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex.

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Disclaimer: The question was given incomplete on the portal. Here is the complete question.

Question: Parabola A can be represented using the equation (x + 3)2 = y, while line B can be represented using the equation y = mx + 9. Isabel claims one solution to the system of two equations must always be the vertex of parabola A. Which best describes the reasonableness of her claim?

The equation in which b varies directly as the square root of c is b = 50 · √c. (Correct choice: B)

In this problem we have a case of direct variation between two variables, which is mathematically described by a direct proportionality model, whose form and characteristics are shown below:

b ∝ √c

b = k · √c      (1)

Where k is the proportionality constant.

First, we determine the value of the constant of proportionality by substituting on b and c and clearing the variable: (b = 100, c = 4)

k = b / √c

k = 100 / √4

k = 100 / 2

k = 50

Then, the equation in which b varies directly as the square root of c is b = 50 · √c. (Correct choice: B)

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Answer:

b

Step-by-step explanation:

using the recursive rule [tex]a_{n}[/tex] = 3[tex]a_{n-1}[/tex] - 1 and a₁ = 2 , then

a₂ = 3a₁ - 1 = (3 × 2) - 1 = 6 - 1 = 5

a₃ = 3a₂ - 1 = (3 × 5) - 1 = 15 - 1 = 14

a₄ = 3a₃ - 1 = (3 × 14) - 1 = 42 - 1 = 41

a₅ = 3a₄ - 1 = (3 × 41) - 1 = 123 - 1 = 122

The surface area of a rectangular prism with a height of 16 feet, a width of 10 feet, and a length of 13 feet is 996 ft².

A = Surface area

l = Length

w = Width

h = Height

A = 2 ( w l + h l + h w )

= 2 · ( 10 · 13 + 16 · 13 + 16 · 10 )

= 996 ft²

The gap occupied by using a dimensional flat surface is referred to as the region. it's far measured in rectangular gadgets. The location occupied by using a 3-dimensional object by means of its outer floor is referred to as the surface location.

The surface vicinity of a strong object is a degree of the total vicinity that the surface of the item occupies.

The floor location is the overall location included via all of the faces of a 3-d item. As an example, if we need to discover the amount of paint required to paint a cube, then the surface on which the paint will be implemented is its floor place. it is continually measured in square gadgets.

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The windows width in the drawing according to the task content is; 9.6inches.

According to the task content, the window's actual width on the house is given as; 48 inches.

Hence, since, the window's width on the drawing is 20% of the actual width, it follows that;

Windows drawing width = 20% × 48 = 9.6 inches.

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The best explanation for why the rotation is isometric is "The rectangle did not change shape or size". Hence the second option is the right choice.

There are four main categories of transformations:

A stiff transformation known as an isometry maintains perimeter and area while also preserving length and angle measurements. In other words, there is congruence between the preimage and the image. Translations, reflections, and rotations are therefore isometric, but dilations are not since the image and preimage are comparable, rather than congruent figures.

The transformation in the question will be isometric when the preimage of the rectangle before 360° rotation, will be congruent to the image after the rotation.

The congruency is best described by the option "The rectangle did not change shape or size", as that is the basis of congruency.

Thus, the best explanation for why the rotation is isometric is "The rectangle did not change shape or size". Hence the second option is the right choice.

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The provided question is incomplete. For the complete question, refer to the attachment.

Using compound interest, it is found that they must deposit $9,143.47 in order to have the desired amount.

The amount of money earned, in compound interest, after t years, is given by:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which:

For this problem, the parameters are:

A(15) = 30000, r = 0.08, n = 4

Then we solve for P to find the initial deposit:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

[tex]30000 = P\left(1 + \frac{0.08}{4}\right)^{4 \times 15}[/tex]

[tex]P = \frac{30000}{(1.02)^{60}}[/tex]

P = $9,143.47.

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Answer:

Khan Academy.

Step-by-step explanation:

You can go on khan academy and search up "percentages and numbers." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.

The center of the circle = (8, -4)

The radius of the circle =  [tex]\sqrt{76}[/tex]

The given equation of the circle is:

[tex]x^2+y^2-16x+8y+4=0[/tex]

The equation can be expressed and simplified as:

[tex]x^2-16x+y^2+8y=-4\\\\x^2-16y+8^2+y^2+8y+4^2=-4+8^2+4^2\\\\(x-8)^2+(y+4)^2=-4+64+16\\\\(x-8)^2+(y+4)^2=76[/tex]

The general equation of a circle is:

[tex](x-a)^2+(y-b)^2=r^2[/tex]

Comparing the two equations:

The center, (a, b) = (8, -4)

The radius, r = [tex]\sqrt{76}[/tex]

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It is determined while using the binomial distribution that there is still a 1.145=114.5% chance that she produces no more than 11 of them.

There are just two possible results for each throw. Either she succeeds or she fails. The binomial probability distribution is employed to answer this issue since the probability of completing a shot is regardless of all other throws.

Binomial probability distribution-

[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]

[tex]C_{n,x} = n!/x! (n-x)![/tex]

where,

the no. of success= x

the no. of trials = n

the probability of a success on one trial = p

The probability of throwing not more than 11 will be:

P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)

Where,

[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]

[tex]P(X=0) = C_{18,0} .(0.689)^{0}(0.311)^{18}[/tex]≈0

[tex]P(X=1) = C_{18,1} .(0.689)^{1}(0.311)^{17}[/tex]≈0

[tex]P(X=2) = C_{18,2} .(0.689)^{2}(0.311)^{16}[/tex]≈0

[tex]P(X=3) = C_{18,3} .(0.689)^{3}(0.311)^{15}[/tex]≈0

[tex]P(X=4) = C_{18,4} .(0.689)^{4}(0.311)^{14}[/tex]≈0

[tex]P(X=5) = C_{18,5} .(0.689)^{5}(0.311)^{13}[/tex]= 0.0003

[tex]P(X=6) = C_{18,4} .(0.689)^{6}(0.311)^{12}[/tex]=0.0016

[tex]P(X=7) = C_{18,7} .(0.689)^{7}(0.311)^{11}[/tex]=0.0062

[tex]P(X=8) = C_{18,8} .(0.689)^{8}(0.311)^{10}[/tex]=0.0188

[tex]P(X=9) = C_{18,9} .(0.689)^{9}(0.311)^{9}[/tex]=0.0463

[tex]P(X=10) = C_{18,10} .(0.689)^{10}(0.311)^{8}[/tex]=0.9232

[tex]P(X=11) = C_{18,11} .(0.689)^{11}(0.311)^{7}[/tex]=0.1488

So,

P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)

=0+0+0+0+0+0.0003+0.0016+0.0062+0.0188+0.0463+0.9232+0.1488 =1.145

Therefore, she makes 1.145=114.5% probability, no more than 11 of them.

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See below for the solution to each question

Yes, the graph is a function.

This is because all x values have different y values

This is the set of input values of the graph.

From the graph, we have

x = 0 to x = 17

Hence, the domain is [0, 17]

This is the set of output values of the graph.

From the graph, we have

y = 0 to y = 10

Hence, the range is [0, 10]

This is the maximum point on the graph.

From the graph, we have

Maximum = (12, 10)

This is the minimum point on the graph.

From the graph, we have

Minimum = (0, 0)

These are the intervals where the y values increase as x increase.

From the graph, we have

Increasing intervals = (0, 5) ∪ (10, 12) ∪ (14, 15)

These are the intervals where the y values decrease as x increase.

From the graph, we have

Decreasing intervals = (7, 10) ∪ (12, 14) ∪ (15, 17)

These are the intervals where the y values remain unchanged as x changes.

From the graph, we have

Constant intervals = (5, 7)

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Answer:

Area of parallelogram = 9600 m²

Step-by-step explanation:

• We can find the measure of angle x using the cos rule:

[tex]160^2 = 100^2 +100^2 -2(100)(100) \cdot cos (x)[/tex]

Make x the subject of the equation:

⇒  [tex]20000 \cdot cos(x) = 100^2 +100^2 - 160^2[/tex]

⇒  [tex]cos(x) = -0.28[/tex]

⇒  [tex]x = cos^{-1} (-0.28)[/tex]

⇒  [tex]x = 106.26 \textdegree[/tex]

• Now we can find the area of one the triangles formed using the formula:

[tex]Area =\frac{1}{2} ab \cdot sin \theta[/tex]

where a and b are two sides of a triangle, and θ is the angle between them (angle x).

Substituting the values:

Area of one triangle = [tex]\frac{1}{2}[/tex]  × (100)(100) × sin(106.26°)

                             = 4800 m²

• Since the parallelogram is formed by two such triangles, we have to double the area of the triangle to find the parallelogram's area:

Area of parallelogram = 2 × 4800

                                     = 9600 m²