Luther bought a snack pack of crackers, and he thinks it's interesting that the cylindrical
tower of crackers came in a box shaped like a square prism. He wonders how much extra
space the box contains around the cracker tower. The box has a height of 8 inches and a side
length of 2 inches. If the tower of crackers has a height of 7.9 inches and a diameter of
1.9 inches, what is the volume of the extra space in the box?
Round your answer to the nearest tenth.

Answer:

1)  695.3 m²

2)  8 ft

3)  172.0 in²

Step-by-step explanation:

To find the area of a regular polygon, we can use the following formula:

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the side length.\\\end{minipage}}[/tex]

Given the polygon is an octagon, n = 8.

Given each side measures 12 m, s = 12.

Substitute the values of n and s into the formula for area and solve for A:

[tex]\implies A=\dfrac{(12)^2 \cdot 8}{4 \tan\left(\dfrac{180^{\circ}}{8}\right)}[/tex]

[tex]\implies A=\dfrac{144 \cdot 8}{4 \tan\left(22.5^{\circ}\right)}[/tex]

[tex]\implies A=\dfrac{1152}{4 \tan\left(22.5^{\circ}\right)}[/tex]

[tex]\implies A=\dfrac{288}{\tan\left(22.5^{\circ}\right)}[/tex]

[tex]\implies A=695.29350...[/tex]

Therefore, the area of a regular octagon with side length 12 m is 695.3 m² rounded to the nearest tenth.

[tex]\hrulefill[/tex]

The sum of an interior angle of a regular polygon and its corresponding exterior angle is always 180°.

If the exterior angle of a polygon measures 40°, then its interior angle measures 140°.

To determine the number of sides of the regular polygon given its interior angle, we can use this formula, where n is the number of sides:

[tex]\boxed{\textsf{Interior angle of a regular polygon} = \dfrac{180^{\circ}(n-2)}{n}}[/tex]

Therefore:

[tex]\implies 140^{\circ}=\dfrac{180^{\circ}(n-2)}{n}[/tex]

[tex]\implies 140^{\circ}n=180^{\circ}n - 360^{\circ}[/tex]

[tex]\implies 40^{\circ}n=360^{\circ}[/tex]

[tex]\implies n=\dfrac{360^{\circ}}{40^{\circ}}[/tex]

[tex]\implies n=9[/tex]

Therefore, the regular polygon has 9 sides.

To determine the length of each side, divide the given perimeter by the number of sides:

[tex]\implies \sf Side\;length=\dfrac{Perimeter}{\textsf{$n$}}[/tex]

[tex]\implies \sf Side \;length=\dfrac{72}{9}[/tex]

[tex]\implies \sf Side \;length=8\;ft[/tex]

Therefore, the length of each side of the regular polygon is 8 ft.

[tex]\hrulefill[/tex]

The area of a regular polygon can be calculated using the following formula:

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the side length.\\\end{minipage}}[/tex]

A regular pentagon has 5 sides, so n = 5.

If its perimeter is 50 inches, then the length of one side is 10 inches, so s = 10.

Substitute the values of s and n into the formula and solve for A:

[tex]\implies A=\dfrac{(10)^2 \cdot 5}{4 \tan\left(\dfrac{180^{\circ}}{5}\right)}[/tex]

[tex]\implies A=\dfrac{100 \cdot 5}{4 \tan\left(36^{\circ}\right)}[/tex]

[tex]\implies A=\dfrac{500}{4 \tan\left(36^{\circ}\right)}[/tex]

[tex]\implies A=\dfrac{125}{\tan\left(36^{\circ}\right)}[/tex]

[tex]\implies A=172.047740...[/tex]

Therefore, the area of a regular pentagon with perimeter 50 inches is 172.0 in² rounded to the nearest tenth.

Answer:

1.695.29 m^2

2.8 feet

3. 172.0477 in^2

Step-by-step explanation:

1. The area of a regular octagon can be found using the formula:

[tex]\boxed{\bold{Area = 2a^2(1 + \sqrt{2})}}[/tex]

where a is the length of one side of the octagon.

In this case, a = 12 m, so the area is:

[tex]\bold{Area = 2(12 m)^2(1 + \sqrt{2}) = 288m^2(1 + \sqrt2)=695.29 m^2}[/tex]

Therefore, the Area of a regular octagon is 695.29 m^2

2.

The formula for the exterior angle of a regular polygon is:

[tex]\boxed{\bold{Exterior \:angle = \frac{360^o}{n}}}[/tex]

where n is the number of sides in the polygon.

In this case, the exterior angle is 40°, so we can set up the following equation:

[tex]\bold{40^o=\frac{ 360^0 }{n}}[/tex]

[tex]n=\frac{360}{40}=9[/tex]

Therefore, the polygon has n=9 sides.

Perimeter=72ft.

We have

[tex]\boxed{\bold{Perimeter = n*s}}[/tex]

where n is the number of sides in the polygon and s is the length of one side.

Substituting Value.

72 feet = 9*s

[tex]\bold{s =\frac{ 72 \:feet }{ 9}}[/tex]

s = 8 feet

Therefore, the length of each side of the polygon is 8 feet.

3.

Solution:

A regular pentagon has five sides of equal length. If the perimeter of the pentagon is 50 in, then each side has a length = [tex]\bold{\frac{perimeter}{n}=\frac{50}{5 }= 10 in.}[/tex]

The area of a regular pentagon can be found using the following formula:

[tex]\boxed{\bold{Area = \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *s^2}}[/tex]

where s is the length of one side of the Pentagon.

In this case, s = 10 in, so the area is:

[tex]\bold{Area= \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *10^2=172.0477 in^2}[/tex]

Drawing: Attachment

Answer:

d)  171.8°

e)  72

f)  60°

Step-by-step explanation:

The Polygon Interior Angle Theorem states that measure of the interior angle of a regular polygon with n sides is [(n - 2) · 180°] / 2.

The number of sides of a 44-gon is n = 44. Therefore, the measure of its interior angle is:

[tex]\begin{aligned}\textsf{Interior angles of a 44-agon}&=\dfrac{(44-2) \cdot 180^{\circ}}{44}\\\\&=\dfrac{42\cdot 180^{\circ}}{44}\\\\&=\dfrac{7560^{\circ}}{44}\\\\&=171.8^{\circ}\;\sf(nearest\;tenth)\end{aligned}[/tex]

Therefore, the interior angle of a 44-gon is 171.8°.

[tex]\hrulefill[/tex]

The Polygon Interior Angle Theorem states that measure of the interior angle of a regular polygon with n sides is [(n - 2) · 180°] / 2.

Given the interior angle of a regular polygon is 175°, then:

[tex]\begin{aligned} \textsf{Interior angle}&=175^{\circ}\\\\\implies \dfrac{(n-2) \cdot 180^{\circ}}{n}&=175^{\circ}\\\\(n-2)\cdot 180^{\circ}&=175^{\circ}n\\\\180^{\circ}n-360^{\circ}&=175^{\circ}n\\\\5^{\circ}n&=360^{\circ}\\\\n&=72\end{aligned}[/tex]

Therefore, the number of sides of the regular polygon is 72.

[tex]\hrulefill[/tex]

According the the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles of a polygon is 360°.

Therefore, to find exterior angle of a regular hexagon, divide 360° by the number of sides:

[tex]\begin{aligned}\sf Exterior\;angle&=\dfrac{360^{\circ}}{\sf Number\;of\;sides}\\\\&=\dfrac{360^{\circ}}{6}\\\\&=60^{\circ}\end{aligned}[/tex]

Therefore, the exterior angle of a regular hexagon is 60°.

Answer:

4. a. 171.818°

b. 72 sides

c. 60°

Step-by-step explanation:

d. The interior angle of a regular polygon with n sides can be calculated using the formula:

[tex]\bold{Interior Angle =\frac{ (n-2) * 180\°}{n}}[/tex]

For a regular 44-gon, the interior angle would be:

[tex]\bold{Interior Angle =\frac{ (44-2) * 180\°}{44}=171.818^o}[/tex]

[tex]\hrulefill[/tex]

e. The number of sides in a regular polygon with an interior angle of 175° can be found using the formula:

[tex]\bold{n = \frac{360\° }{180\° - Interior \:Angle}}[/tex]

For an interior angle of 175°, the number of sides would be:

substituting Value,

[tex]\bold{n =\frac{ 360\° }{ 180\° - 175\°} = \frac{360\° }{ 5\° }= 72 \:sides}[/tex]

[tex]\hrulefill[/tex]

f.

The sum of the exterior angles of any polygon is always 360°.

Since a regular hexagon has six sides,
n=6

exterior angle would be [tex]\bold{\frac{360\° }{ n}}[/tex]

substituting value,

exterior angle=[tex]\bold{\frac{360\° }{ 6}=60\°}[/tex]

[tex]\hrulefill[/tex]

The length of B'C' from the given quadrilateral A'B'C'D' is 3 units. Therefore, option D is the correct answer.

A translation in math moves a shape left or right and/or up or down. The translated shapes look exactly the same size as the original shape, and hence the shapes are congruent to each other. They just have been shifted in one or more directions.

Here, AB = A'B' = 7 units

AD = A'D'= 6 units

DC = D'C'= 4 units

BC = B'C' = 3 units

Therefore, option D is the correct answer.

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Answer:

The GCF (Greatest Common Factor) of two numbers is the largest number that divides both of them. In this case, the GCF of 36 and 60 is 12.

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The coordinates of point M are (1, 2).

The gradient of CB is 5.

The equation of line AD in the form y = mx + c is: y = (-1/5)x + 37/5.

To solve the given problem, we can follow these steps:

1. Calculate the coordinates of point M:

Since the diagonals of a rhombus bisect each other, the midpoint of the diagonal AC will give us the coordinates of point M.

Midpoint formula:

x-coordinate of M = (x-coordinate of A + x-coordinate of C) / 2

                 = (-3 + 5) / 2

                 = 2 / 2

                 = 1

y-coordinate of M = (y-coordinate of A + y-coordinate of C) / 2

                 = (8 - 4) / 2

                 = 4 / 2

                 = 2

Therefore, the coordinates of point M are (1, 2).

2. The gradient (slope) of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the formula:

Gradient (m) = (-4 - 1) / (5 - 6)

            = -5 / -1

            = 5

Therefore, the gradient of CB is 5.

3. To find the equation of line AD, we need to calculate the gradient (m) of AD and the y-intercept (c).

Gradient of CB = 5

Gradient of AD = -1/5 (negative reciprocal of 5)

To find the y-intercept (c), we can substitute the coordinates of point A (-3, 8) into the equation y = mx + c and solve for c:

8 = (-1/5)(-3) + c

8 = 3/5 + c

c = 8 - 3/5

c = 40/5 - 3/5

c = 37/5

Therefore, the equation of line AD in the form y = mx + c is:

y = (-1/5)x + 37/5.

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Answer:

To calculate Mr. Chi's total tax bill, we need to find the amount of sales tax charged by each entity and add them together.

The state sales tax on the sale of a house is 3.2%, which means Mr. Chi will owe 0.032 x $289,300 = $9,267.20 in state sales tax.

The county sales tax on the sale of a house is 2.6%, which means Mr. Chi will owe 0.026 x $289,300 = $7,520.80 in county sales tax.

The city sales tax on the sale of a house is 1.8%, which means Mr. Chi will owe 0.018 x $289,300 = $5,207.40 in city sales tax.

Therefore, Mr. Chi's total tax bill will be $9,267.20 + $7,520.80 + $5,207.40 = $21,995.40.

Answer: C

Step-by-step explanation:

If I plugged in 0 for x,  You get 4 for y.   This is the y-intercept

(0,4)

The only graph that goes through (0,4) is C.

The statement "The cup of flour has a mass of exactly 121.8 grams" is not true.

Since the baking scale measures mass to the tenth of a gram, it means it can measure values such as 121.7 grams, 121.8 grams, and 121.9 grams. However, it does not have the capability to measure beyond the tenths place, so it cannot determine whether the exact mass of the cup of flour is exactly 121.8 grams. Therefore, the statement claiming an exact mass is not true.

The other statements are all true:

[tex]\huge{\mathcal{\colorbox{black}{\textcolor{lime}{\textsf{I hope this helps !}}}}}[/tex]

♥️ [tex]\large{\textcolor{red}{\underline{\texttt{SUMIT ROY (:}}}}[/tex]

Answer:

Step-by-step explanation:

A baking scale measures mass to the tenth of a gram, up to 650 grams. A cup of flour is placed on the scale and results in a measure of 121.8 grams. [ Which of the following statements is not true? a.The exact mass of the cup of flour must be between 121.7 and 121.9 grams.

keeping in mind that the sum of all exterior angles in a polygon always add up to 360°, then

[tex]58+39+50+48+59+x+x~~ = ~~360\implies 254+2x=360 \\\\\\ 2x=106\implies x=\cfrac{106}{2}\implies x=53[/tex]

Answer:

X is 53 because 106 divide by 2 is equal to 53

Answer:

[tex] {x}^{ - \frac{5}{3} } = \frac{1}{ {x}^{ \frac{5}{3} } } = \frac{1}{ \sqrt[3]{ {x}^{5} } } [/tex]

Answer:

a) .12 × $2,400 = $288

b) $2,400 + $288 = $2,688

Answer: Factor the problem

Step-by-step explanation:

Take out the common factor (-16), it will leave you with (t^2 - t - 30)

Factor (t - 6) (t + 5)

t - 6 = 0 or t + 5 = 0

t = 6 seconds

The calculated value of the missing side length in the triangle is (c) 12

From the question, we have the following parameters that can be used in our computation:

The simiar triangles

Using the theorem of corresponding sides, we hav

?/8 = 9/6

Multiply both sides by 8

So, we have

? = 8 * 9/6

Evaluate

? = 12

Hence, the missing side length in the triangle is (c) 12

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The volume of the cylinder with a height of 12 cm and a diameter of 8 cm is approximately 602.19 cm³.

To calculate the volume of a cylinder, you can use the formula:

Volume = π * (radius²) * height

Given that the diameter of the cylinder is 8 cm, the radius (r) can be calculated by dividing the diameter by 2:

radius (r) = 8 cm / 2 = 4 cm

The height (h) of the cylinder is 12 cm.

substitute these values into the formula:

Volume = π * (4 cm)² * 12 cm

Volume = π * 16 cm² * 12 cm

Volume ≈ 602.19 cm³ (rounded to two decimal places)

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Answer:

---------------------

The segment GI contains both GH and HI.

Using the segment addition postulate we set up an equation:

Substitute to get:

So the value of x is 9.

Answer:

Step-by-step explanation:

For the given function:

Vertex is at (-2, -4) and range is −4 ≤ y < ∞

Hence, Option b is correct.

The given function is

y = |3x + 6| − 4

We can see that it is consist of absolute value function or mod function.

Since we know that,

An absolute value function is an algebraic function in which the variable is contained inside the absolute value bars.

The absolute value function is also known as the modulus function, and its most frequent form is f(x) = |x|,

where x is a real integer. In general, the absolute value function may be represented as f(x) = a |x - h| + k,

where a denotes how far the graph extends vertically, h represents the horizontal shift, and k represents the vertical displacement from the graph of f(x) = |x|.

If the value of 'a' is negative, the graph opens downwards; otherwise, it opens upwards.

The appropriate method of finding range and vertex both is to plot its graph:

Therefore after plotting graph we get,

Vertex is at (-2, -4)

And range is (-4 , ∞) ⇒ −4 ≤ y < ∞

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Answer:

See below for answers and explanations

Step-by-step explanation:

a, b, and c are the coefficients of the quadratic, so a=1, b=6, and c=-12

(h,k) is the vertex of the parabola, so we must convert it to vertex form by completing the square:

[tex]y=x^2+6x-12\\y+21=x^2+6x-12+21\\y+21=x^2+6x+9\\y+21=(x+3)^2\\y=(x+3)^2-21[/tex]

Therefore, the vertex is (3,-21), which means h=-3 and k=-21.

The y-intercept is when x=0, or when the function passes through the x-axis:

[tex]y=x^2+6x-12\\y=0^2+6(0)-12\\y=-12[/tex]

Therefore, the y-intercept is -12 (also written as (0,-12) as an ordered pair).

The x-intercept is when y=0, or when the function passes through the y-axis:

[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x=\frac{-6\pm\sqrt{6^2-4(1)(-12)}}{2(1)}\\\\x=\frac{-6\pm\sqrt{36+48}}{2}\\\\x=\frac{-6\pm\sqrt{94}}{2}[/tex]

Therefore, the x-intercepts are (-6+√94)/2 and (-6-√94)/2.

Hope this helped!

The quadrant of the terminal side for 15) is fourth quadrant.

To change from radians to degrees, you need to the multiple the number of radians by 180/π.

Multiple the number of radians by 180/π.

15) 5 radians = 5 * (180/π)

16) 11π/6 = (11*180)/6 = 330.

19) 3.5 = 3.5*(π/180) degree.

20) 1 = 1*(π/180) degree.

therefore, terminal side for 15) is fourth quadrant.

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The function is a good model for the data.

The retail sales in the U.S. in 2016 is estimated to be 282.24 billion dollars.

The year that corresponds to retail sales of $241 billion is 2020.

The function F(t)= 9.44 t +84.182 is a linear model for the data. This means that the graph of the function is a straight line.

The scatter plot shows that the data points are close to a straight line. This means that the function is a good model for the data.

To estimate the retail sales in the U.S. in 2016, we need to find the value of F(11). F(11) = 9.44 * 11 + 84.182

= 282.24.

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Step-by-step explanation:

points on the line: (4, -2) & (-5, 3)

gradient of the line = -5/9

general equation for all straight lines: y = mx + c

substitute one coordinate and the gradient into the equation. 3 = (-5/9)(-5) + c

therefore, c = 2/9

so the general equation is y = (-5/9)x + 2/9

Use tracing paper for the rotation. Put a dot on the origin (0,0) and trace the shape. Then turn the paper 180 degrees and redraw the shape in its new position.

For the reflection, the line y = x + 2 would have an x interrcept of (-2,0) and y intercept of (0,2) so, draw a line through those coordinates and reflect the new shape you traced. That will be your final answer, you can rub out the rotated one as the question asked you to rotate AND reflect. Not just rotate.

Answer:

664

Step-by-step explanation:

6x17

6x17

10x6

10x6

10x17

10x17

add them all up and u get 664

Answer:

The remaining Jugs will take us 18 minutes to fill.

Explenation:

14 -2 =

12 jugs remain

2 = 3min /2

1 = 1.5min

12 x 1.5= 18min

The area of the composite figures are:

The total area of Shape A is = 115.5 in²

and, total area of Shape B is = 51ft².

Here, we have,

from the given diagram we get,

Shape A.)

Part 1: it is a rectangle,

l= 13 and w = 7

so, area = 91 in²

Part 2: it is a triangle,

b = 7 and h = 7

so, area = 1/2 *7 * 7 = 24.5 in²

So, The total area of Shape A is = 115.5 in²

now, Shape B.)

part 1 : it is a rectangle,

l=4 and w = 6

so, area = 24 ft²

part2: it is a square,

side = 3

so, area = 9 ft²

part 3: it is a trapezium,

a = 6, b = 3, h =4

so, area = 1/2 * (6+3) * 4 = 18ft²

So, total area of Shape B is = 51ft².

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the measure of the angle within the 10 degrees rotation is determined as 170 degrees.

The measure of the angle within the 10 degrees rotation is calculated by applying sum of circle theorem as shown  below.

The sum of angles in the straight line 180 degrees, and we can apply this principle in calculating the expected angle within the 10 degrees rotation as follows;

Let the expected measure of the angle = θ

θ + 10 = 180 ( sum of angles in a straight line )

θ = 180 - 10

θ = 170

Thus, the measure of the angle within the 10 degrees rotation is determined as 170 degrees.

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Based on the given information, the average annual salary for all US teachers is $48,000 with a standard deviation of $5,700. So, you should feel good about your $52,000 offer, as it is above the average salary for US teachers and places you in a relatively higher earning position compared to your peers.

If you apply for a teaching position and were offered $52,000, let's determine your relative position in the salary distribution using a z-score.

The z-score is calculated as (X - μ) / σ, where X is your salary, μ is the mean salary, and σ is the standard deviation. Plugging in the values, we have:
Z = ($52,000 - $48,000) / $5,700 = $4,000 / $5,700 ≈ 0.70
A z-score of 0.70 indicates that your salary is 0.70 standard deviations above the mean salary. Based on a normal distribution table, about 24% of teachers would earn more than you, and 76% would earn less.

Considering the probability result, you should feel good about your $52,000 offer, as it is above the average salary for US teachers and places you in a relatively higher earning position compared to your peers.

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Answer:

50

Step-by-step explanation:

The angles of a triangle add up to 180°

So

53+77+x=180

x=180-130

x=50

Set B_V spans V and is linearly independent, satisfying the requirements for a basis.

The statement is true.

If W is a subspace of a vector space V and B_W is a basis for W, then it is true that there exists a unique subspace U such that V = W ⊕ U, where ⊕ represents the direct sum of subspaces.

This means that every vector in V can be uniquely expressed as the sum of a vector in W and a vector in U.

Additionally, there exists a basis B_U for U such that the union of B_W and B_U, denoted as B_V = B_W ∪ B_U, forms a basis for the entire vector space V.

This means that the set B_V spans V and is linearly independent, satisfying the requirements for a basis.

Hence, the statement is true.

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The triangle with points at (2, 4), (4, 2), (9, 6) was reflected across the line y = 3 to get the points (2, 2), (4, 4), (9, 0)

Transformation is the movement of a point from its initial point to a new location. Types of transformation are reflection, rotation, translation and dilation.

The triangle with points at (2, 4), (4, 2), (9, 6) was reflected across the line y = 3 to get the points (2, 2), (4, 4), (9, 0)

Thus, option (D) is correct.

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