What is the product of 5/6 and 2/3?

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:

So you would do 1/3 times 4 so that is 1 and 1/3

The total distance that was run by Runners who each ran for 1/3 of a mile​ will be 4/3.

The distance is defined as the length of the space between the two points separated from each other.

From the graph, the distance of 1/3 miles is covered by 4 runners so the total distance will be calculated as:-

Distance = 1/3  x  4

Distance = 4/3  miles

Hence the total distance that was run by Runners who each ran for 1/3 of a mile​ will be 4/3.

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[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:

-2,-4

Step-by-step explanation:

Answer:

-2, -4

Step-by-step explanation:

   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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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:

  square inch — is a unit of area, not volume

Step-by-step explanation:

Some volume measures have their own unit, like liters, cups, pints, quarts, gallons, bushels, or barrels.

Other volume units are the cube of a linear measure, such as cubic centimeters, cubic inches, cubic feet, or cubic meters.

Still other volume units are a hybrid of an area measure and a linear measure, such as acre-feet.

A measure that is the square of a linear unit is an area measure, not a volume measure.

Answer:

Step-by-step explanation:

"square inch" is a unit of area, not of volume.

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

Answer:

Step-by-step explanation:

if the temp is -8°F and you add 19 your new temp is 11°F

Final answer:

To find the final temperature after a rise of 19°F from -8°F, define the variable T as the final temperature, and use the expression T = initial temperature + temperature change, which evaluates to T = 11°F.

Explanation:

The student is asked to determine the temperature after a rise of 19°F from an initial temperature of -8°F. To solve this, let's define the variable T to represent the final temperature. The expression to find the final temperature after the rise would be:

T = initial temperature + temperature change

In this case, the initial temperature is -8°F and the temperature change is a rise of 19°F, so the expression becomes:

T = -8°F + 19°F

When we evaluate this expression:

T = 11°F

Therefore, the temperature after a rise of 19°F from -8°F is 11°F.

Final answer:

The student performed relatively better on the Geography test than on the Mathematics test, with z-scores of -1.2 and -1.5 respectively, because a higher z-score indicates a performance closer to the mean.

Explanation:

To determine on which test the student scored better relative to the other students, we must calculate the z-scores for each test.

The z-score formula is:

Z = (X - μ) /σ

Where X is the student's score, μ is the mean, and σ is the standard deviation.

For the Geography test:

Z = (56 - 80) / 20
Z = -24 / 20
Z = -1.2

For the Mathematics test:

Z = (285 - 300) / 10
Z = -15 / 10
Z = -1.5

A z-score tells us how many standard deviations an element is from the mean. The student had a z-score of -1.2 in Geography and -1.5 in Mathematics. Since a higher z-score reflects a performance that is closer to the mean, the student performed relatively better on the Geography test compared to the Mathematics test.

Answer: well i can see 34 eyeballs in the jar, if you multipy that by pie, and then divide by 2 then multiplyb b to. the cut it into third by the mcd, (most common deenominator) you get 4216

Step-by-step explanation: i ran these numbers for weeks on weeks, please tell me if it worked i haveen't slept in days.

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

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:

a. (-3,2)

b. sqrt65

c.

Step-by-step explanation:

a. To find the center of the circle, you can think of it just like finding the midpoint between the two endpoints. To find a midpoint between two endpoints, you take the average of the x values to get the x coordinate, and you take the average of the y values to get the y coordinate of the midpoint. Therefore, if (-10, -2) is (x1, y1) and (4, 6) is (x2, y2), the midpoint/center of the circle would be:

( (x1+x2)/2, (y1+y2)/2 ). When you plug in our x and y values, you get (-3, 2).

b. To find the radius of a circle, you need to know the center/midpoint of the circle which we solved for in part a. The formula for finding the radius of a circle with the center is (x-h)^2 + (y-k)^2 = r^2 for (h, k) as the center. The coordinates of the center that we found earlier for this circle are (-3, 2). With that, we just plug in our numbers into the formula, and we get:

(x+3)^2 + (y-2)^2 = r^2. Now, to get r, we can choose one of the original two endpoints given and plug in the x and y coordinates from that point into this equation. I like (4, 6), so I'm going to plug in 4 for x and 6 for y, and so we get (4+3)^2 + (6-2)^2 = r^2 which equals 49 + 16 = r^2 when simplified. 49 plus 16 is equal to 65, so we get 65 = r^2. To finally get r, we square root both sides of the equation to get r =  sqrt65 which is already in the simplest radical form.

c. The circle equation is (x-h)^2 + (y-k)^2 = r^2, like I said in part b. Therefore, we already have our circle equation! We just plug in our center points and we get (x+3)^2 + (y-2)^2 = sqrt65. This is usually an equation a question will give you for a circle, and with this information, they will expect you to find the center (h, k) or the circle and it's radius, r.

The center of the circle with diameter endpoints P(-10, -2) and Q(4, 6) is C = (-3, 2). The radius in radical form is √65. The equation for the circle is (x + 3)² + (y - 2)² = 65.

Finding the Center and Radius of a Circle

To find the center of the circle, you need to find the midpoint of the diameter with endpoints P(-10, -2) and Q(4, 6). The midpoint formula is:

((x1 + x2) / 2, (y1 + y2) / 2)

Plugging in our points, we get:

Center (C) = ((-10 + 4) / 2, (-2 + 6) / 2) = (-3, 2)

Finding the Radius in Radical Form

To find the radius of the circle, we calculate the distance between P and Q and then divide it by 2. The distance formula is:

√[(x2 - x1)² + (y2 - y1)²]

The distance (diameter) is √[(4 - (-10))² + (6 - (-2))²] = √[14² + 8²] = √[196 + 64] = √260. Therefore, the radius (r) is √260/2 or √65.

The Equation of the Circle

The standard form equation for a circle with a center (h, k) and radius r is (x - h)² + (y - k)² = r². Therefore, the equation for this circle, with center (-3, 2) and radius √65, is:

(x + 3)² + (y - 2)² = 65

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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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

___

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

___

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.

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Answer:

(2,1) is the only solution to the linear equation

Step-by-step explanation:

to solve, just plug in the values given into the orignal expression. remember, the answer on the left has to be equal to the answer on the right.

(2,1)

4(2) + 3(1) = 11 --> 8 + 3 = 11 --> 11 = 11 (2,1) is a solution to the linear equation

(6,0)

4(6) + 3(0) = 11 --> 24 + 0 = 11 --> 24 ≠ 11 (6,0) is not a solution to the linear equation

(0,1)

4(0) + 3(1) = 11 --> 0 + 3 = 11 --> 3 ≠ 11 (0,1) is not a solution to the linear equation

Answer:

86.86

Step-by-step explanation:

Another word for mean is average.  To find the mean or average, we add up all the numbers and then divide by the number of numbers

There are 7 scores

Mean = (88+91+89+94+90+64+ 92)/7

         = 608/7

         =86.85714286

To 2 decimal places ( we have to round to be accurate to 2 decimal places)

         86.86

Answer:

[tex]b = 60 {(1.05)}^{3} \\ b = 69.46[/tex]

B = $69.46

Step-by-step explanation:

It should be B = 69.46 but lower case was the only option given for variables.

Answer:

If the landscaper decides not to purchase the tallest plant, then the median heights of the plants would (increase-decrease-stay the same) , but the mean height would (increase-decrease-stay the same)

Step-by-step explanation:

The Complete sentences are:

The median heights of the plants would stay the same.

The mean height would decrease.

Based on the values of each point, a dot plot visually groups the number of data points in a data set. Similar to a histogram or probability distribution function, this provides a visual representation of the data distribution.

We have,

A dot plot shows the heights of the plants a landscaper plans to purchase.

The median heights of the plants would stay the same.

If the landscaper decided not to buy the tallest plant, however the mean height would decrease.

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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]

(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:

F(x) < 0 , over the intervals (-∞, -0.7) and (0.76 , 2.5)

Step-by-step explanation:

The correct answer is the first option

F(x) < 0 , over the intervals (-∞, -0.7) and (0.76 , 2.5)

We can see that over those intervals, the function becomes negative.

That is, it becomes less than zero.

We can see the graph in the image below.

Answer:

93.6 inch^2

Step-by-step explanation:

sides=6 side-length=6 inches Apothem= 5.2 inches Area= 6*(1/2)(6)(5.2)

The area of the regular hexagon is 93.6 square centimeter.

Area is a collection of two - dimensional points enclosed by a single dimensional line. Mathematically, we can write area as -

V = ∫∫F(x, y) dx dy

Given is a regular hexagon with a side length of 6 cm and an apothem of approximately 5.2.

We can write the area of the regular hexagon as -

A = {3√3}/2 x a²

A = {3√3}/2 x 6 x 6

A = 93.6 square centimeter

Therefore, the area of the regular hexagon is 93.6 square centimeter.

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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:

   4 yd 2 ft 5 in

Step-by-step explanation:

One straightforward way to do this is to convert to inches and back:

  28 yd 2 ft 6 in = (28 yd)(36 in/yd) + (2 ft)(12 in/ft) + 6 in

  = 1008 in + 24 in + 6 in = 1038 in

Then your division problem looks like ...

  (1038 in)/6 = 173 in

Converting this back to yards, feet, and inches can proceed like this ...

  (173 in)/(36 in/yd) = 4 yd remainder 29 in . . . . . find the number of yards first

  = 4 yd + (29 in)/(12 in/ft) = 4 yd 2 ft 5 in . . . . . then find feet and inches

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.

[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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Answer:

6

Step-by-step explanation:

Cost per item is found by dividing the cost by the number of items. If the woman bought n items for $120, the cost of each item is $120/n. If the woman bought 24 more items, n+24, at the same price, then the cost per item is $120/(n+24). The problem statement tells us this last cost is $16 less than the first cost:

120/(n+24) = (120/n) -16

Multiplying by n(n+24) gives ...

120n = 120(n+24) -16(n)(n+24)

0 = 120·24 -16n^2 -16·24n . . . . . . subtract 120n and collect terms

n^2 +24n -180 = 0 . . . . . . . . . . . . . divide by -16 to make the numbers smaller

(n +30)(n -6) = 0 . . . . . . . . . . . . . . factor the quadratic

The solutions to this are the values of n that make the factors zero: n = -30, n = 6. The negative value of n has no meaning in this context, so n=6 is the solution to the equation.

The woman bought 6 items.

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Check

When the woman bought 6 items for $120, she paid $120/6 = $20 for each of them. If she bought 6+24 = 30 items for the same money, she would pay $120/30 = $4 for each item. That amount, $4, is $16 less than the $20 she paid for each item.

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

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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

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

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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.

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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.