What is the point of maximum growth rate? Round to the nearest tenth.

Answer:

D)

Step-by-step explanation:

Answer:

9.78 u² . . . . . best matches selection B

Step-by-step explanation:

The formula for the area of a segment is ...

A = (1/2)r²(θ-sin(θ)) . . . . . θ is the central angle in radians, r is the radius

Filling in the given values, we have ...

A = (1/2)·(6√3)²(π/3 -sin(π/3)) = 54(π/3 -√3/2) ≈ 9.7832960 u²

_____

Inappropriate rounding of intermediate computation values will give a different result. The value of π/3-sin(π/3) is the small difference of relatively larger numbers, so minor errors in the values of those numbers can have a great effect on the difference.

Answer:

Option a.

I and III

Step-by-step explanation:

Observing the graph

For n=400 coats

The cost is about $2,200

and

The revenue is less than $1,400

Substitute the value of n=400 in each equation to find the solution

I Cost 1.5(400)+1,600=$2,200 ----> is ok

II Cost 4.5(400)+1,600=$3,400 ----> is not ok ( is greater than $2,200)

III Revenue 3.25(400)=$1,300 ----> is ok ( is less than $1,400)

IV Revenue 5.75(400)=$2,300 ----> is not ok ( is greater than $1,400)

therefore

The solution is I and III

Answer:

A

Step-by-step explanation:

Answer:

the answer is D

Step-by-step

14ab^3/7a^-2b^-1

= 14a^3b^4/7

=2a^3b^4

Answer:

  226 ft²

Step-by-step explanation:

The surface of the figure can be considered in several parts:

a) top and bottom surfaces

b) outside surfaces

c) inside surfaces

__

The top and bottom surfaces each consist of a rectangle 8 ft by 4 ft with a 4 ft by 1 ft hole. The area of the larger rectangle without the hole is the product of its length and width:

  (8 ft)(4 ft) = 32 ft²

The area of the hole is the product of its length and width:

  (4 ft)(1 ft) = 4 ft²

Then the area of the surface around the hole is the difference of these:

  32 ft² - 4 ft² = 28 ft²

The top and bottom surfaces together have twice this area for a total of ...

  top and bottom area = 2·(28 ft²) = 56 ft²

__

The outside (lateral) area is the total area of the four rectangles that make up the sides of the figure. Each rectangle has a height of 5 ft, so we can compute the area by finding the perimeter of the figure and multiplying that by 5 ft.

The perimeter is the sum of the lengths of its top or bottom edges:

  8 ft + 4 ft + 8 ft + 4 ft = 2·(8 ft +4 ft) = 2·12 ft = 24 ft

Then the lateral area is ...

  outside lateral area = (5 ft)(24 ft) = 120 ft²

__

The area of the sides of the hole can be computed the same way. The hole is 5 ft high and its edge lengths are 1 ft and 4 ft. Then the total inside lateral area is ...

  inside lateral area = (5 ft)(2·(1 ft + 4ft)) = 50 ft²

__

So the total surface area of the solid is ...

  total area = top and bottom area + outside lateral area + inside lateral area

  total area = (56 + 120 + 50) ft²

  total area = 226 ft²

By Stokes' theorem, the line integral of [tex]\vec F[/tex] over [tex]C[/tex] is equivalent to the surface integral of the curl of [tex]\vec F[/tex] over [tex]S[/tex], where [tex]S[/tex] is the part of the plane [tex]7x+6y+z=1[/tex] in the first octant, with [tex]S[/tex] having positive/upward orientation.

Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=\dfrac{(1-u)(1-v)}7\,\vec\imath+\dfrac{u(1-v)}6\,\vec\jmath+v\,\vec k[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex].

Take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k[/tex]

The curl of [tex]\vec F[/tex] is

[tex]\nabla\times\vec F(x,y,z)=(x-y)\,\vec\imath-y\,\vec\jmath+\vec k[/tex]

Then the line integral is equivalent to

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]

[tex]=\displaystyle\int_0^1\int_0^1\left(\frac{(6-13u)(1-v)}{42}\,\vec\imath-\dfrac{u(1-v)}6\,\vec\jmath+\vec k\right)\cdot\left(\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k\right)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle\frac1{252}\int_0^1\int_0^1(12-6v-19u+19uv)(1-v)\,\mathrm du\,\mathrm dv=\boxed{\frac{11}{1512}}[/tex]

First, find the normal vector of the plane, and then find the curl of the vector field to obtain a new vector field to be integrated over the given surface. Then perform the surface integral of this resultant vector field over the surface to get the required output.

Using Stokes' Theorem, we first find the normal vector of the plane 7x + 6y + z = 1 by extracting the coefficients, i.e., the vector is (7,6,1). Next, find the curl of the vector field F(x, y, z) = i + (x + yz)j + (xy - sqrt(z ))k. This provides the vector field, let's say G, that you are going to integrate over the surface that C bounds. The curl is calculated as follows:

The curl of F = ∇ x F= ( ∂/∂y [(xy - sqrt(z))] - ∂/∂z [(x + yz)] )i - ∂/∂x [(xy - sqrt(z))]j + ∂/∂x [(x + yz)]k.

Then perform the surface integral of this resultant vector field, G, over the surface bounded by C to get your answer. The surface integral is ∫∫ G·dS.

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Answer:

option D

2x4; -11

Step-by-step explanation:

Order of matrix is in form (m x n), here m is the row and n is the column of the matrix.

So this matrix have 2 rows and 4 columns

1)Order of matrix

2x4

2)[tex]a_{12}[/tex]

here 1 is the row and 2 is the column

-11

Each elements of the matrix can be identity as below

[tex]\left[\begin{array}{cccc}x_{11} &x_{12} &x_{13} &x_{14} \\x_{21} &x_{22} &x_{23} &x_{24} \\\end{array}\right][/tex]

[tex]\displaystyle\\\text{We have the points:}\\E(5,~8)\\F(12,~0)\\G(-5,~0)\\H(-12,~8)\\\\EF=\sqrt{(5-12)^2+(8-0)^2}=\sqrt{(-7)^2+8^2}=\\=\sqrt{49+64}=\sqrt{113}\approx\boxed{\bf10.63}\\\\FG=12-(-5)=12+5=\boxed{\bf17}\\\\GH=\sqrt{(-5-(-12))^2+(0-8)^2}=\sqrt{7^2+(-8)^2}=\\=\sqrt{49+64}=\sqrt{113}\approx\boxed{\bf10.63}\\\\HE=|-12-5|=|-17|=\boxed{\bf17}\\\\P=EF+FG+GH+HE=10.63+17+10.63+17=55.26\approx\boxed{\bf55.3}\\\\\text{Correct answer: }~B)[/tex]

Answer:

[tex]\pi\dfrac{12.1}{3}\left(\dfrac{12.1}{\tan 57^{\circ}}\right)^2[/tex]

Step-by-step explanation:

You are interested in two of the sides of the right triangle, the leg AC and the leg BC. The trig function that relates their values to the angle shown is told you by the mnemonic SOH CAH TOA, which reminds you ...

  Tan = Opposite/Adjacent

The side opposite the angle, BC is given as 12.1; the side adjacent is AC, designated r. Then the above relation tells you ...

  tan(57°) = 12.1/r

Rearranging, we have ...

  r = 12.1/tan(57°)

The volume of the cone is given by the formula ...

  V = (1/3)πr²h . . . . . where h = 12.1

Filling in what we know, this is ...

  V = (1/3)π(12.1/tan(57°))²·12.1

This can be rearranged to the form shown in your answer choices:

  V = π(12.1/3)(12.1/tan(57°))² . . . . . . matches the lower-right choice

Answer:

a.

Step-by-step explanation:

Answer:

  see the attachment

Step-by-step explanation:

• When two secants intersect a circle the measure of the angle between them is half the difference of the intercepted arcs. Tangent DH is a degenerate case of a secant in which both points of intersection are the same point (D). Thus ∠GHD is half the difference of arc GD and ED.

• We assume points B and C are on radii DA and EA, respectively. Hence arc DE has the same measure as ∠BAC, and ∠DGE has half that measure.

• Tangent DH will intersect radius DA at right angles. Hence ∠ADH = 90°.

• ∠DHE = ∠DHG, whose measure is given by the formula in the first answer choice. The formula in the 4th choice is different (and wrong).

• ∠DGE and ∠BFC both intercept arcs with the same measure. Neither is greater than the other: they are equal in measure.

• Inscribed triangle DGE cannot be a right triangle because none of its sides is a diameter of the circumcircle.

Answer:

ghd bac adh

Step-by-step explanation:

ANSWER

[tex]\frac{41\pi}{90} [/tex]

EXPLANATION

To find how many degrees are in 82°,

We convert 82° to radians.

To do that we multiply by

[tex] \frac{\pi}{180 \degree} [/tex]

This implies that,

[tex]82 \degree =82 \degree \times \frac{\pi}{180 \degree} = \frac{41\pi}{90} [/tex]

Hence there are

[tex] \frac{41\pi}{90} [/tex]

radians in 82 degrees.

Answer is lots of god Which equation is the inverse of y = 100 – x2?









The answer should be cos2x

Answer:

[tex]\cos^2x[/tex]

Step-by-step explanation:

The given expression is:

[tex](\cos x)(\sec x)-\sin^2 x[/tex]

Recall form Pythagorean Identity that;

[tex]\sec x=\frac{1}{\cos x}[/tex]

We apply this property to obtain;

[tex](\cos x)(\frac{1}{\cos x})-\sin^2 x[/tex]

We simplify to get;

[tex]1-\sin^2 x[/tex]

Recall from the Pythagorean identity that;

[tex]1-\sin^2 x=\cos^2x[/tex]

Answer:

25.

Step-by-step explanation:

44:11*100 = 25

Answer:

25%

Step-by-step explanation:

Of means multiply and is means equals

P *44 =11

Divide each side by 44

P *44/44=11/44

P = 1/4

P = .25

Change from decimal form to percent form by multiplying by 100

P = 25%

Answer:

(B) [tex]SA=72{\tex{feet^2}[/tex]

Step-by-step explanation:

Given: From the figure, it is given that the length of the base is 4 feet and slant height is 7 feet.

To find: The Surface are of Pyramid.

Solution: From the figure, it is given that the length of the base is 4 feet and slant height is 7 feet.

Now, surface area of the Pyramid is given as:

[tex]SA={\text{Area of base}+\frac{1}{2}pl[/tex]

where p is the perimeter and l is the slant height.

Now, area of base is given as:

[tex]A=4(4){\tex{ft^2}[/tex]

[tex]A=16{\tex{ft^2}[/tex]

And, the surface area is given as:

[tex]SA=16+\frac{1}{2}(4)(4)(7)[/tex]

[tex]SA=16+56[/tex]

[tex]SA=72{\tex{feet^2}[/tex]

Hence, option  B is correct.

Answer:

B 72 ft^2

Step-by-step explanation:

The area surface of a square pyramid is given by adding the area of the square that creates the base, and then the area of the 4 triangles that make up for the sides of the pyramid, so we first calculate the area of the triangle:

Area= b*h/2

Area= 4*7/2

Area=14

Now we calculate the area of the base:

Area=side*side

Area=4*4

Area=16

No we add up the four triangles plus the base:

Surface area=(Sides*4)+base

Surface area= (14*4)+16

Surface area=56+16

Surface area=72

So the surface area of the pyramid would be 72 ft^2

Answer:

6. C: {x^2 +(y-1)^2 =2; x+y = 3}

15. C: The line does not intersect the circle.

Step-by-step explanation:

The formula for the distance (d) from a point (x, y) to a line ax+by=c is ...

d = |ax+by-c|/√(a^2+b^2)

The formula for a circle centered at (h, k) with radius r is ...

(x -h)^2 +(y -k)^2 = r^2

___

6. Comparing the circle equation to the generic equation, we find (h, k) = (0, 1) and r = √2. Then we want to find the line that is distance √2 from the center of the circle. Our line equation is x+y=c for some value of c that we want to find.

d = √2 = |0 +1 -c|/√(1^2+1^2)

2 = |1-c|

±2 = 1-c

c = 1±2 = -1 or 3

The line that is tangent to the circle is the one of choice C: x+y = 3

__

The attached graph shows the lines for all 4 answer choices. The point of tangency is (1, 2), so x+y=1+2=3.

___

15. The circle is centered at (4, 1) and has radius 3. The distance from the circle center to the line is ...

d = |2(4) -(1)|/√(2^2+(-1)^2) = 7/√5 ≈ 3.13

The distance from the circle center to the line is more than the radius of the circle, so there can be no points of intersection.

__

Alternate solution

You can substitute for y using the equation of the line. Then the circle equation becomes ...

(x -4)^2 + (2x -1)^2 = 9

x^2 -8x +16 +4x^2 -4x +1 = 9

5x^2 -12x +8 = 0

The discriminant of this quadratic is ...

b^2 -4ac = (-12)^2 -4(5)(8) = 144-160 = -16

Since this value is negative, there can be no real solutions, meaning the line does not intersect the circle.

Answer:

⁵/₁₂

Step-by-step explanation:

1. The probability or rolling a number > 1 (given all sides have equal chances):

Possible outcomes: 1, 2, 3, 4, 5, 6; Wanted outcomes: 2, 3, 4, 5, 6

5/6 of the possible outcomes are wanted.

2. Chance of flipping heads (given both sides have an equal chance):

Possible outcomes: heads, tails; Wanted outcomes: heads

1/2 of the possible outcomes are wanted.

3. Probability of both: ⁵/₆ * ¹/₂ = ⁵/₁₂

Answer:

Step-by-step explanation:

The 9th term is the next-to-last term, where the left term of the binomial is raised to the first power, the right term of the binomial is raised to the 8th power (9-1=8), and the multiplier is 9C8 = 9!/(8!·1!) = 9. This product is ...

  9·(3x)^1·(-2y)^8 = 6912xy^8

Answer:

Step-by-step explanation:

Interest earned is proportional to the interest rate, so if the interest earned is the same, the amounts invested must be inversely proportional to the interest rates. That is, for the 3% and 2% accounts, the ratio of money invested is 2:3.

In other words, 2/5 of the money ($3200) was invested at 3%, and 3/5 of the money ($4800) was invested at 2%.

_____

If you need an equation, you can let x represent the amount invested at the highest rate. Then 8000-x is the amount invested at the lower rate. For the interest in the two accounts to be equal, we have ...

  3%·x = 2%·(8000-x) . . . . . the amounts of interest earned are the same

  3/2·x = 8000 -x . . . . . . . . divide by 2%

  5/2·x = 8000 . . . . . . . . . . . add x and simplify

  x = 8000·(2/5) = 3200  . . . multiply by the inverse of the x coefficient

  8000-x = 8000 -3200 = 4800 . . . . the amount invested at the lower rate

Tamara invested $3200 in the 3% account and $4800 in the 2% account.

_____

She earned $96 in each account for the year.

Answer:

  the appropriate choice is the last one

Step-by-step explanation:

[tex]7^{\frac{4}{5}}=(7^4)^{\frac{1}{5}}=\sqrt[5]{7^4}[/tex]

This is equivalent to b^(pq) with b=7, p=4, q=1/5. Here, q is a rational number, as suggested by the last answer choice.

Answer:

The correct options are:

So, these functions have the same range.

The functions have the same base.

The functions have the same domain.

g(x) is a translation left 1 unit.

Step-by-step explanation:

According to the table, both are exponential functions.

We have that

[tex]f(x) = 3^{x}[/tex]

[tex]g(x) = 3^{x+1}[/tex]

Lets see each affirmation:

The functions have the same base.

An exponential function [tex]a^{x}[/tex] has base a.

In this problems, both f and g have base 3.

The functions have the same range.

The range of f are all the values that f can assume. That is, all the positive numbers.

The range of g are all the values that g can assume. That is, also all the positive numbers.

So, these functions have the same range.

The functions have the same exponent.

An exponential function [tex]a^{x}[/tex] has exponent x.

f has exponent x and g has exponent x + 1. So those functions do not have the same exponent.

The functions have the same domain.

Yes, they both have x = {0,1,2,3} as domain.

g(x) is a translation left 1 unit.

g(x) = f(x+1). So yes, g(x) is a translation left 1 unit.

g(x) is a translation right 2 units.

g(x) is not f(x-2). So g(x) is not a translation right 2 units.

g(x) is a translation up 2 units.

g(x) is not f(x) + 2. So g(x) is not a translation up 2 units.

Answer:

A B D E

Step-by-step explanation:

Just did it!

   Image points of A, B, C and D by reflecting them about x-axis will be A'(1, -3), B'(-2, 2), C'(-4, -5) and D'(2, 5).

        P(h, k) → P'(h, -k)

Following the rule for the reflection,

Image points by reflecting the points A, B, C and D will be,

A(1, 3) → A'(1, -3)

B(-2, -2) → B'(-2, 2)

C(-4, 5) → C'(-4, -5)

D(2, -5) → D'(2, 5)

        Therefore, image points of A, B, C and D by reflecting them about x-axis will be A'(1, -3), B'(-2, 2), C'(-4, -5) and D'(2, 5).

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90 degrees you are looking to your side

180 degrees you are looking behind you

around origin of 0,0

the image is flipped into the negative world if it is in posiitve or vice versa

The coordinates of the triangle after a rotation of 180° counterclockwise is given by P' ( -3 , 2 ) , Q' ( -8 , 2 ) , R' ( -5 , 5 )

The measure of the amount a figure is rotated about the center of rotation is called the angle of rotation. The angle of rotation is usually measured in degrees. We specify the degree measure and direction of a rotation.

90° clockwise rotation: (x,y) becomes (y,-x)

90° counterclockwise rotation: (x,y) becomes (-y,x)

180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

270° clockwise rotation: (x,y) becomes (-y,x)

270° counterclockwise rotation: (x,y) becomes (y,-x)

Given data ,

Let the rotation angle be represented as A

Now , the value of A is A = 180° counterclockwise

Let the triangle be represented as PQR

Now , the coordinates of the triangle PQR is

The coordinate of P = P ( 3 , -2 )

The coordinate of Q = Q ( 8 , -2 )

The coordinate of R = R ( 5 , -5 )

And , when 180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

Substituting the values in the equation , we get

The coordinate of point P' = P' ( -3 , 2 )

The coordinate of point Q' = Q' ( -8 , 2 )

The coordinate of point R' = R' ( -5 , 5 )

Hence , the coordinates of the triangle after rotation is

P' ( -3 , 2 ) , Q' ( -8 , 2 ) , R' ( -5 , 5 )

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(a) The bond pays 5% of its face value each year. That amount is ...

  0.05 × $14,000 = $700 . . . . per year

So, in 6 years, the bond pays ...

  6 × $700 = $4200

__

(b) The bond continues to pay $700 per year for the next 24 years. At the end of that time, the face value of the bond is also paid. So, the total amount paid to the bondholder in 24 years is ...

  24 × $700 + 14,000 = $16,800 +14,000 = $30,800

__

(c) The present value of the bond is the present value of the cash flows it will generate. It will pay $700 at the end of each year for 24 years, then $14,000 at the end of the 24th year. Assuming cash flows are discounted at 7.5% per year over that period, the present value will be ...

  present value of series of payments = (700/1.075)·(1.075^-24 -1)/(1.075^-1 -1)

  ... = 7688.08

  present value of final payment = 14,000·1.075^-24 = 2467.88

So, the total present value is ...

  present value of the bond = $7,688.08 +2,467.88 = $10,155.96

Answer:

  B)  24/25

Step-by-step explanation:

You're asked to find the exact value of the composition ...

  sin(2·arccos(3/5))

You know the double angle formula for the sine is ...

  sin(2x) = 2sin(x)cos(x)

and you know the sine can be derived from the cosine by ...

  sin(x) = √(1 -cos(x)^2)

Since you're given the cosine as 3/5, this means you need to find the value of ...

  sin(2·arccos(3/5)) = 2·√(1 -(3/5)^2)·(3/5) = 2·(√(25-9))/5·3/5 = 2·4/5·3/5

  sin(2·arccos(3/5)) = 24/25 . . . . . . matches the marked answer

_____

If you do this on your calculator, you will find the result to be 0.96, which is the decimal value of 24/25.

Answer:

-2,-4

Step-by-step explanation:

Answer:

-2, -4

Step-by-step explanation:

[tex]f(x)=\begin{cases}5x-8&\text{for }x<0\\|-4-x|&\text{for }x\ge0\end{cases}[/tex]

The limits from either side are

[tex]\displaystyle\lim_{x\to0^-}f(x)=\lim_{x\to0}(5x-8)=-8[/tex]

[tex]\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0}|-4-x|=\lim_{x\to0}|x+4|=|4|=4[/tex]

The one-sided limits don't match, so the limit as [tex]x\to0[/tex] does not exist.

A function assigns the values. The limit as x→0 does not exist.

A function assigns the value of each element of one set to the other specific element of another set.

The given limits can be written as,

[tex]f(x)=\left \{{5x-8\ \ \ {\rm for\ x < 0} \atop |-4-x|\ \ \ {\rm for\ x \geq 0}} \right.[/tex]

Now, the limits from either side are,

[tex]\lim_{x \to 0^-} f(x) = \lim_{x \to 0} (5x-8) = -8[/tex]

[tex]\lim_{x \to 0^+} f(x) = \lim_{x \to 0} |-4-x| = \lim_{x \to 0} |x+4|=|4| = 4[/tex]

Since one side of the limits doesn't match, so the limit as x→0 does not exist.

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The answers are:

First image:

The answer is the second option, the angles is [tex]53\°[/tex]

Second image:

The answer is the third option:

[tex]\frac{5}{13}[/tex]

Third image:

The length of the adjacent leg is the first option:

[tex]8\sqrt{2}units[/tex]

Fourth image:

The answer is the fourth option, [tex]72\°[/tex]

Fifth image:

The answer is the fourth option, DF (hypothenuse) is equal to 25 units.

To solve these problems, we need to use the following trigonometric identities and the Pythagorean Theorem, since we are working with right triangles.

[tex]Tan(\alpha)=\frac{y}{x}\\\\(Tan(\alpha))^{-1} =(\frac{y}{x})^{-1}\\\\\alpha =Arctan(\frac{y}{x})[/tex]

[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]

Pythagorean Theorem:

[tex]c^{2}=a^{2} +b^{2}[/tex]

So, solving we have:

First image:

We are given a right triangle that has the following lengths:

[tex]base=x=6units\\height=y=8units\\hypothenuse=10units[/tex]

Then, calculating we have:

[tex]\alpha =Arctan(\frac{y}{x})\\\\\alpha =Arctan(\frac{8}{6})\\\\\alpha =Arctan(1.33)\\\\\alpha =53\°[/tex]

Hence, the answer is the second option, the angles is [tex]53\°[/tex]

Second image:

We are given a right triangle that has the following lengths:

[tex]base=x=12units\\height=y=5units\\hypothenuse=13units[/tex]

Then calculating the sin ratio, we have:

[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]

[tex]Sin(\alpha)=\frac{5}{13}[/tex]

Thence, the answer is the third option:

[tex]\frac{5}{13}[/tex]

Third Image:

We are given the following information:

[tex]hypothenuse=16units\\\\\alpha =45\°[/tex]

Then, calculating one of the angle legs, since both will have the same length, using the sine trigonometric identity, we have:

[tex]Sin(\alpha)=\frac{Opposite}{Hypothenuse}\\ \\Sin(45\°)=\frac{Opposite}{16}\\\\Opposite=Sin(45\°)*16\\\\Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}[/tex]

Hence, the answer is the first option the length of the adjacent leg is

[tex]Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}units[/tex]

Fourth image:

We are given the following information:

[tex]base=x=9units\\height=y=3units[/tex]

To calculate the angle at the B vertex, first, we need to calculate the angle at the C vertex, and then, calculate the B vertex by the following way:

Since the sum of all the interior angles of a triangle are equal to 180°, we have that:

[tex]180\°=Angle_{B}+Angle{C}+90\°[/tex]

[tex]Angle_{B}=180\° -90\°-Angle_{C}[/tex]

So, calculating the angle at the C vertex, we have:

[tex]\alpha =Arctan(\frac{y}{x})[/tex]

[tex]\alpha =Arctan(\frac{3}{9})[/tex]

[tex]\alpha =Arctan(0.33)=18.26\°[/tex]

Then, calculating the angle at the B vertex, we have:

[tex]Angle_{B}=180\° -90\°-18.26\°=71.74\°=71.8\°=72\°[/tex]

Hence, the answer is the fourth option, [tex]72\°[/tex]

Fifth image:

We are given the following information:

[tex]base=x=24units\\height=y=7units[/tex]

Now, to calculate the distance DF (hypothenuse) we need to use the Pythagorean Theorem:

[tex]c^{2}=a^{2} +b^{2} \\\\hypothenuse^{2}=adjacent^{2}+opposite^{2}\\\\\sqrt{hypothenuse^{2}}=\sqrt{adjacent^{2}+opposite^{2}}\\\\hypothenuse=\sqrt{adjacent^{2}+opposite^{2}}[/tex]

Then, substituting we have:

[tex]hypothenuse=\sqrt{24^{2}+(7)^{2}}[/tex]

[tex]hypothenuse=\sqrt{576+49}=\sqrt{625}[/tex]

[tex]hypothenuse=\sqrt{625}[/tex]

[tex]hypothenuse=25units[/tex]

Hence, the answer is the fourth option, DF (hypothenuse) is equal to 25 units.

Have a nice day!

Answer:

Step-by-step explanation:

There are a few formulas that are useful for this:

You will notice that for lateral area purposes, a pyramid or cone is equivalent to a prism or cylinder of height equal to half the slant height. And for volume purposes, the volume of a pyramid or cone is equal to the volume of a prism or cylinder with the same base area and 1/3 the height.

Since the measurements are given in cm, we will use cm for linear dimensions, cm^2 for area, and cm^3 for volume.

___

The heights of the cones at the top of the towers can be found from the Pythagorean theorem.

  (slant height)^2 = (height)^2 + (radius)^2

  height = √((slant height)^2 - (radius)^2) = √(10^2 -5^2) = √75 = 5√3

The heights of the pyramids can be found the same way.

  height = √(13^2 -2^2) = √165

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Area

The total area of the castle will be ...

  total castle area = castle lateral area + castle base area

These pieces of the total area are made up of sums of their own:

  castle lateral area = cone lateral area + pyramid lateral area + cylinder lateral area + cutout prism lateral area

and ...

  castle base area = cylinder base area + cutout prism base area

So, the pieces of area we need to find are ...

Here we go ...

Based on the above discussion, we can add 1/2 the slant height of the cone to the height of the cylinder and figure the lateral area of both at once:

  area of one cone and cylinder = π·10·(18 +10/2) = 230π

  area of cylinder with no cone = top area + lateral area = π·1^2 +π·2·16 = 33π

  area of one pyramid = 4·4·(13/2) = 52

The cutout prism outside face area is equivalent to the product of its base perimeter and its height, less the area of the rectangular cutouts at the top of the front and back, plus the area of the inside faces (both vertical and horizontal).

  outside face area = 2((23+4)·11 -3·(23-8)) = 2(297 -45) = 504

  inside face area = (3 +(23-8) +3)·4 = 84

So the lateral area of the castle is ...

  castle lateral area = 2(230π + 52) +33π + 504 + 84 = 493π +692

  ≈ 2240.805 . . . . cm^2

The castle base area is the area of the 23×4 rectangle plus the areas of the three cylinder bases:

  cylinder base area = 2(π·5^2) + π·1^2 = 51π

  prism base area = 23·4 = 92

  castle base area = 51π + 92 ≈ 252.221 . . . . cm^2

Total castle area = (2240.805 +252.221) cm^2 ≈ 2493.0 cm^2

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Volume

The total castle volume will be ...

  total castle volume = castle cylinder volume + castle cone volume + castle pyramid volume + cutout prism volume

As we discussed above, we can combine the cone and cylinder volumes by using 1/3 the height of the cone.

  volume of one castle cylinder and cone = π(5^2)(18 + (5√3)/3)

  = 450π +125π/√3 ≈ 1640.442 . . . . cm^3

 volume of flat-top cylinder = π·1^2·16 = 16π ≈ 50.265 . . . . cm^3

The volume of one pyramid is ...

  (1/2)4^2·√165 = 8√165 ≈ 102.762 . . . . cm^3

The volume of the entire (non-cut-out) castle prism is the product of its base area and height:

  non-cutout prism volume = (23·4)·11 = 1012 . . . . cm^3

The volume of the cutout is similarly the product of its dimensions:

  cutout volume = (23 -8)·4·3 = 180 . . . . cm^3

so, the volume of the cutout prism is ...

  cutout prism volume = non-cutout prism volume - cutout volume

  = 1012 -180 = 832 . . . .  cm^3

Then the total castle volume is ...

  total castle volume = 2·(volume of one cylinder and cone) + (volume of flat-top cylinder) +2·(volume of one pyramid) +(cutout prism volume)

  = 2(1640.442) + 50.265 +2(102.762) +832 ≈ 4368.7 . . . . cm^3

The total area of the castle is approximately 2493.026 square centimeters, and the total volume is around 4368.7 cubic centimeters.

The total area of the castle is the sum of its lateral and base areas. The lateral area is composed of the lateral areas of two cones, two pyramids, one cylinder, and the lateral area of a cutout prism. The base area consists of the bases of three cylinders and the base of the cutout prism.

For the lateral area:

Cone and Cylinder: The lateral area of one cone and cylinder combined is 230π square centimeters.

Flat-Top Cylinder: The area of the cylinder with a flat top is 33π square centimeters.

Pyramid: The lateral area of one pyramid is 52 square centimeters.

Cutout Prism: The outside face area is 504 square centimeters, and the inside face area is 84 square centimeters.

The total lateral area of the castle is 493π + 692 square centimeters, which is approximately 2240.805 square centimeters.

For the base area:

Cylinder Bases: The combined area of the bases of the three cylinders is 51π square centimeters.

Prism Base: The base area of the rectangular prism is 92 square centimeters.

The total base area of the castle is 51π + 92, approximately 252.221 square centimeters.

Therefore, the total area of the castle is the sum of the total lateral area and the total base area, which is approximately 2240.805 + 252.221 square centimeters, totaling around 2493.026 square centimeters.

For the volume:

Cylinder and Cone: The volume of one cylinder and cone is 450π + 125π/√3, approximately 1640.442 cubic centimeters.

Flat-Top Cylinder: The volume of the cylinder with a flat top is 16π, approximately 50.265 cubic centimeters.

Pyramid: The volume of one pyramid is 8√165, approximately 102.762 cubic centimeters.

Prism (Non-Cutout): The volume of the non-cutout prism is 23 × 4 × 11, equal to 1012 cubic centimeters.

Cutout Prism: The volume of the cutout prism is 1012 - 180, which is 832 cubic centimeters.

The total volume of the castle is the sum of the volumes of the components, totaling approximately 4368.7 cubic centimeters.

To learn more about area

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Answer:

$665.36

Step-by-step explanation:

2^16 or...

2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 = 66536 pennies