If one side of the triangle is 12 and the opposite angle is 30 degrees, then the diameter of the circumscribed circle is:________

Answer:

Brayden Solved 5 math problem and read 10 pages.

Step-by-step explanation:

Let the number of Math Problem solved be x.

Also Let the Number of Pages he read be y.

Given:

The number of pages Brayden read is twice the number of math problems he solved.

Hence equation can be framed as;

[tex]y =2x[/tex]

Also Given:

Ayden can solve each math problem in 3 minutes and he can read each page in 1 minute and it took him 50 minutes to complete all of his homework.

Hence the equation can be framed as;

[tex]4x+2.5y =45[/tex]

Now Substituting the value of y in above equation we get;

[tex]4x+2.5(2x) =45\\\\4x+5x=45\\\\9x=45\\\\x=\frac{45}{9}=5[/tex]

Now Substituting the value of x to find the value of y we get;

[tex]y =2x =2\times5 =10[/tex]

Hence Brayden Solved 5 math problem and read 10 pages.

Answer:

67 squares or 66.66 squares.

2/3 turned into a decimal is 66.66 or rounded, 67 squares.

Hope this helped! (plz mark me brainliest!)

Answer:

  A(2, 2)

Step-by-step explanation:

I find it useful to graph the given points. The center of rotation is at the place where the perpendicular bisectors of BB' and CC' meet. That point is ...

  A =  (2, 2).

__

The graph shows you the slope of BB' is 1, so its perpendicular bisector will have a slope of -1. The midpoint of BB' is (2-2, 6+2)/2 = (0, 4). This is the y-intercept of the line, so the perpendicular bisector of BB' has equation ...

  y = -x +4

The slope of CC' is -1/3, so its perpendicular will have a slope of -1/(-1/3) = 3. The midpoint of CC' is (4+1, 3+4)/2 = (5/2, 7/2). In point-slope form the equation of the perpendicular bisector of CC' is ...

  y -7/2 = 3(x -5/2)

  2y -7 = 3(2x -5) . . . . multiply by 2

  2(-x+4) -7 = 3(2x -5) . . . substitute for y

  -2x +1 = 6x -15 . . . . .eliminate parentheses

  16 = 8x . . . . . . . . . . add 2x+15

  x = 2 . . . . . . divide by 8

  y = -2+4 = 2 . . . . from the equation for y

The intersection of the perpendicular bisectors of BB' and CC' is the center of rotation:

  A = (2, 2)

Final answer:

The coordinate of vertex A is (2, 2), found by equating B' to (y, x) and using the given B' coordinates.

Explanation:

To find the coordinates of vertex A, we can use the properties of a rotation. A 90° counterclockwise rotation about a point involves switching the coordinates and negating the y-coordinate. Let's denote the coordinates of A as (x, y). After the rotation, A becomes B', so we have:

B' = (y, x)

From the given coordinates of B' (–2, 2), we can equate:

y = 2

-2 = -x

Solving these equations, we find that x = 2. Therefore, the coordinates of vertex A are (2, 2). So, Maggie's image graph depicts a 90° counterclockwise rotation about vertex A (2, 2) of triangle ABC.

Answer:

12

Step-by-step explanation:

Number of students in Bobby's class = 30

40% of the students have blue eyes.

It means 40 out of 100 students will have blue eyes.

Then we have to find how many out of 30 students will have blue eyes.

Let us take the number of blue-eyed students out of 30 to be x.

Then:

[tex]\frac{40}{100}=\frac{x}{30}\\x=\frac{40}{100}\times30\\x=12[/tex].

Hence 12 out of 30 students will have blue eyes.

when you multiply $1.59 by 5 you get $7.95

when you multiply $1.87 by 2 you get $4.74

so when you subtract them you get $3.21

Answer:

length of the pretty side  and length of the side oppositte to the pretty side  =   37.91 ft

length of the other two sides  = 27.52 ft

Step-by-step explanation:

The mathematical problem is:

Max A = b1*h

subject to: 35*b1 + 18*(2*h + b2) <= 3000

Where

A: area of the garden

b1: length of the pretty side

b2: length of the side oppositte to the pretty side

h: length of the other two sides

Replacing with b1 = b2 and taking only the equality sign in the restriction (in the maximum all the money will be spent), we get:

35*b1 + 18*(2*h + b1) = 3000

35*b1 + 36*h + 18*b1 = 3000

53*b1 + 36*h = 3000

b1 = 3000/53 - (36/53)*h

Substituing in Area's formula  

A = (3000/53 - (36/53)*h)*h

A = (3000/53)*h - (36/53)*h^2

In the maximum, the derivative of A is equal to zero

dA/dh = 3000/53 - 2*(36/53)*h =

3000/53 - 72/35*h = 0

h = (3000/53)*(35/72)

h = 27.52 ft

then,

b1 = 3000/53 - (36/53)*27.52

b1 = 37.91 ft =b2

The dimensions of the garden that give Mr. Rogers the maximum area are approximately: Length (L) = 28.3 feet and Width (W) = 41.7 feet.

To find the dimensions of Mr. Rogers's garden that will give him the maximum area while staying within his budget, we need to set up and solve a problem involving optimization with constraints.

First, let's define the variables:

The cost of fencing:

The total cost of fencing can be expressed as follows:

[tex]35L + 18(2W + L) = 3000[/tex]

Simplifying this equation:

[tex]35L + 36W + 18L = 3000[/tex]
[tex]53L + 36W = 3000[/tex]

To find the dimensions that maximize the area, we need to express the area in terms of one variable and use calculus to find the maximum. Let's solve for one variable in terms of the other. We'll solve for [tex]W[/tex]:

[tex]53L + 36W = 3000[/tex]
[tex]36W = 3000 - 53L[/tex]
[tex]W = \frac{3000 - 53L}{36}[/tex]

Now, express the area [tex]A[/tex] as a function of [tex]L[/tex]:

[tex]A = L \cdot W[/tex]
[tex]A = L \left(\frac{3000 - 53L}{36}\right)[/tex]
[tex]A = \frac{3000L - 53L^2}{36}[/tex]

To find the maximum area, we take the derivative of [tex]A[/tex] with respect to [tex]L[/tex] and set it to zero:

[tex]\frac{dA}{dL} = \frac{3000 - 106L}{36}[/tex]

Set the derivative to zero and solve for [tex]L[/tex]:

[tex]\frac{3000 - 106L}{36} = 0[/tex]
[tex]3000 - 106L = 0[/tex]
[tex]106L = 3000[/tex]
[tex]L = \frac{3000}{106}[/tex]
[tex]L \approx 28.3[/tex]

Now we use this value of [tex]L[/tex] to find the corresponding value of [tex]W[/tex]:

[tex]W = \frac{3000 - 53 \times 28.3}{36}[/tex]
[tex]W \approx \frac{3000 - 1499.9}{36}[/tex]
[tex]W \approx \frac{1500.1}{36}[/tex]
[tex]W \approx 41.7[/tex]

Answer:

A) 210

B) 357

C) 267

Step-by-step explanation:

A) Among 7 games, we can first choose 3 wins, and then among remaining 4 games, we can choose 2.

To calculate the possibility, we will use Combination.

[tex]C(7,3)*C(4,2) = 35 *6 = 210[/tex]

B) Player A can get 4 points with the following cases:

4 wins and 3 loses

3 wins, 2 draws and 2 loses

2 wins, 4 draws and 1 lose

1 win and 6 draws

Indeed, these cases matches for Player B too to get 3 points.

So again, we will use Combination to calculate the possibility.

[tex]C(7,4) + C(7,3)*C(4,2) + C(7,2)*C(5,4) + C(7,1) = 357[/tex]

C) Here, we need to find two possibilities after 6 games and add them, while Player A has 3 points and wins the 7th game, and Player A has 3.5 points and draws the 7th game.

[tex][C(6,3)+C(6,2)C(4,2)+C(6,1)C(5,4)+C(6,6)] + [C(6,3)C(3,1)+C(6,2)C(4,3)+C(6,1)] =[20+90+30+1]+[60+60+6]=141+126=267[/tex]

Answer:

The 10% off sale will save Melissa $10.49.

Step-by-step explanation:

Given:

The listed price of the camera is $195.99.

The camera is no sale for 10% off and Melissa has a coupon for 5% off the sales tax is 7%.

Now, to find the money Melissa will save of the 10% off sale.

So, to get the price of camera of the 5% off sale:

[tex](195.99-5\%\ of\ 195.99)[/tex]

[tex]=(195.99-\frac{5}{100}\times 195.99)[/tex]

[tex]=(195.99-9.80)[/tex]

[tex]=186.19[/tex]

Now, adding the sales tax:

[tex]186.19+7\%\ of\ 186.19[/tex]

[tex]=186.19+\frac{7}{100} \times 186.19[/tex]

[tex]=186.19+13.03[/tex]

[tex]=199.22[/tex]

Thus, the price is $199.22.

Now, to get the price of 10% off sale:

[tex]195.99-10\%\ of\ 195.99[/tex]

[tex]=195.99-\frac{10}{100} \times 195.99[/tex]

[tex]=176.39[/tex]

So, adding sales tax:

[tex]176.39+7\%\ of\ 176.39[/tex]

[tex]=176.39+\frac{7}{100} \times 176.39[/tex]

[tex]=188.73[/tex]

Hence, the price is $188.73.

Now, to get the money 10% off sale will save Melissa:

[tex]199.22-188.73[/tex]

[tex]=10.49[/tex]

Therefore, the 10% off sale will save Melissa $10.49.

Final Answer:

The 10% off sale will save Melissa approximately $19.60.

Explantion:

To calculate the savings from the 10% off sale on the listed price of the digital camera, follow these steps:
1. Determine the listed price of the camera. In this case, the listed price is $195.99.
2. Calculate the discount amount by multiplying the listed price by the discount percentage (expressed as a decimal). For a 10% discount, you would convert that percentage to a decimal by dividing by 100:

10% = 10/100 = 0.10.
3. Multiply the listed price by the decimal form of the discount percentage to find the savings:
  Savings = Listed Price × Discount Percentage
  Savings = $195.99 × 0.10
4. Calculate the actual savings:
  Savings = $195.99 × 0.10 = $19.599
5. Since savings are typically represented in dollar format (rounded to two decimal places), we round the savings to the nearest cent:
  Savings ≈ $19.60
So, the 10% off sale will save Melissa approximately $19.60.

Answer:

Prototype

Step-by-step explanation:

Before a product is made, an original model of the product is first formulated. This is called a prototype. It is not simply a part of the design, but a very important part that the process of creating the final product solely depends on it. It is largely described as a highly representative example of a concept or product.

A prototype is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."

In Engineering, a prototype can be defined as a rudimentary (early) model, release, or sample of a particular product, so as to avail the developers and manufacturers an opportunity and ability to test a concept, functionality, or process within it.

For instance, the main purpose of the prototype of a robot is to determine and show whether or not the robot and all of its features are in tandem with the design specifications and market requirements.

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Answer:

t=2.83

Step-by-step explanation:

Linear Approximation Of Functions

The equation of a line is given by

[tex]y=y_o+m(x-x_o)[/tex]

Where m is the slope of the line and [tex](x_o,y_o)[/tex] are the coordinates of a point through which the line goes.

Given a function B(t), we can build an approximate line to model the function near one point. The value of m is the derivative of B in a specific point [tex](t_o, B_o)[/tex]. The equation becomes

[tex]B(t)=B_o+B'(t_o)(t-t_o)[/tex]

Let's collect our data.

[tex]B'(t)=8e^{0.2cost},\ B_o=B(2.2)=4.5[/tex]

Let's find the required values to build the approximate function near [tex]t_0=2.2[/tex]. We evaluate the derivative in 2.2

[tex]B'(2.2)=8e^{0.2cos2.2}=7.11[/tex]

The function can be approximated by

[tex]B(t)=4.5+7.11(t-2.2)[/tex]

Once we have B(t), we are required to find the value of t, such that

[tex]B(t)=9[/tex]

Or equivalently:

[tex]4.5+7.11(t-2.2)=9[/tex]

Rearranging

[tex]\displaystyle t-2.2=\frac{9-4.5}{7.11}[/tex]

Solving for t

[tex]\displaystyle t=\frac{9-4.5}{7.11}+2.2[/tex]

[tex]\boxed{t=2.83}[/tex]

We find the equation of the linear approximation at t=2.2 using B(2.2) and B'(2.2). We then solve this equation for t when the linear approximation equals 9. This gives us the time at which the linear approximation estimates B(t) = 9.

To solve for the time when the linear approximation gives an output of 9, we need to find the equation of the tangent line (i.e., the linear approximation) at t = 2.2. The linear approximation to a function at a particular point is given by the formula L(t) = f(a) + f'(a)*(t-a), where f is the function, a is the point, and f' denotes the derivative of the function.

Given that B(2.2) = 4.5 and B'(t) = 8e0.2cos(t), we can substitute these into the linear approximation formula to get L(t) = 4.5 + 8e0.2cos(2.2)*(t-2.2).

Next, we solve this equation for t when L(t) = 9: 9 = 4.5 + 8e0.2cos(2.2)*(t-2.2). Solving the equation for t gives the time at which the linear approximation estimates B(t) = 9.

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Answer:

  40 mph

Step-by-step explanation:

3 times the speed in mph will be ...

  3 × 20 = 60 . . . . . more than 43

  3 × 40 = 120 . . . . less than 126

The stopping distance in feet is more than 3 times the speed in mph for a speed of 40 mph.

_____

The answer will depend on the units of the ratio. Here, we are apparently to use units of feet per (mile per hour), that is, (ft·h/mi). The answer would be different for distance in meters and speed in km/h, for example.

Answer:

f(x) = 8x⁴-8x²+1

Step-by-step explanation:

I will assume that f(cos θ) = cos(4θ). Otherwise, f would not be a polynomial. lets divide cos(4θ) in an expression depending on cos(θ). We use this properties

cos(4θ) = cos(2 * (2θ) ) = cos²(2θ) - sin²(2θ) = [ cos²(θ)-sin²(θ) ]² - [2cos(θ)sin(θ)]² = [cos²(θ) - ( 1 - cos²(θ) ) ]² - 4cos²(θ)sin²(θ) = [2cos²(θ)-1]² - 4cos²(θ) (1 - cos²(θ) ) = 4 cos⁴(θ) - 4 cos²(θ) + 1 - 4 cos²(θ) + 4 cos⁴(θ) = 8cos⁴(θ) - 8 cos²(θ) + 1

Thus f(cos(θ)) = 8 cos⁴(θ) - 8 cos²(θ) + 1, and, as a result

f(x) = 8x⁴-8x²+1.

The second hand of a clock is 8 cm long what is the distance of the tip of the second hand travel in 10 minutes

Answer:

The distance travelled by tip of second hand in 10 minutes is 502.4 cm

Solution:

Length of second hand = 8 cm long

In one revolution it travels a circumference of  a circle of which radius is 8 cm

The circumference of circle = [tex]2 \pi r[/tex]

[tex]c = 2 \times \pi \times 8 = 16\pi[/tex]

We are aksed to find the distance of the tip of the second hand travel in 10 minutes

Second hand complete 1 revolution in 1 minute.

Therefore, in 10 minutes it revolves 10 revolution,

In 1 revolution tip of second covers = 16π

Hence distance travelled is given as:

[tex]\rightarrow 16 \pi \times10 = 160 \pi[/tex]

We know that π is a constant equals to 3.14

[tex]\rightarrow 160 \pi = 160 \times 3.14 = 502.4[/tex]

Thus the distance travelled by tip of second hand in 10 minutes is 502.4 cm

Answer:

502.4

Step-by-step explanation:

Answer:

Simple interest is calculated using initial principle while compound interest is calculated considering the interest also .

Step-by-step explanation:

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for using his money. The interest, typically expressed as a percentage, can either be compounded or simple .

Simple interest is based on the principal amount , while compound interest is based on the principal amount and the interest that adds onto it in every period and the final principle is used for calculating the interest.

Simple interest is calculated  on the principal amount of a loan and it's easier to find out than compound interest.

The transformation T could be translation, rotation, reflection, or dilation, altering the position, orientation, or size of the triangle.

The transformation T mapping triangle XYZ to triangle X'Y'Z' could be any of the basic rigid transformations in geometry: translation, rotation, reflection, or dilation.

- A translation involves shifting the entire triangle by a certain distance in a certain direction without changing its orientation or shape.

- A rotation rotates the triangle around a fixed point (e.g., the origin or a specific vertex), altering its orientation but maintaining its shape and size.

- A reflection flips the triangle across a line (e.g., the x-axis, y-axis, or an arbitrary line), producing a mirror image with reversed orientation.

- A dilation scales the triangle uniformly, either enlarging or shrinking it while maintaining its shape and proportions.

To determine which transformation T specifically represents, additional information is needed, such as the coordinates of the vertices X, Y, Z and X', Y', Z', or any constraints or properties of the transformation provided. Each type of transformation has distinct characteristics that can be identified through geometric properties and transformations of coordinates.

In parallelogram ABCD, opposite sides AB and DC are both 8 units long, and AD and BC are both 5 units long.

To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent. We can do this by finding the lengths of the opposite side pairs and showing that they are equal.

Let's denote the points as follows:

- A, B, C, D are the vertices of the quadrilateral ABCD.

- AB and DC are opposite sides, and AD and BC are opposite sides.

Given that AB = 8 units and AD = 5 units, we need to find the lengths of DC and BC.

Since ABCD is a parallelogram, opposite sides are congruent. Therefore, DC = AB = 8 units.

To find BC, we know that BC = AD = 5 units.

Thus, we have:

- AB = DC = 8 units

- AD = BC = 5 units

Since both pairs of opposite sides are congruent, quadrilateral ABCD is a parallelogram.

The length of BC is 5 units.

In the given case, The length of BC is 8 units.

To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent.

Let's denote the lengths of the sides as follows: AB, BC, CD, and DA.

 Given that ABCD is a parallelogram, we know that opposite sides are equal in length.

Therefore, we can equate the lengths of AB to CD and BC to DA.

 Let's assume that the length of side AB (and thus CD, since they are opposite and equal) is given as 8 units.

We are asked to find the length of side BC (which will be equal to the length of side DA).

 Since we do not have any additional information such as angles or diagonals, we cannot calculate the length of BC directly.

However, if we are given that ABCD is a parallelogram, then by definition, the lengths of opposite sides are equal.

Therefore, without loss of generality, we can state that the length of BC is also 8 units, which is equal to the length of AB.

 Thus, we have shown that both pairs of opposite sides are congruent:

AB = CD = 8 units

BC = DA

Since AB = 8 units and AB = CD, it follows that CD = 8 units.

Similarly, since AB = CD and AB = 8 units, by the properties of a parallelogram, BC must also be equal to 8 units.

 Therefore, the length of BC is 8 units, confirming that quadrilateral ABCD is indeed a parallelogram.

Answer:

1 in

Step-by-step explanation:

We are given that

Diameter of casting=8 in

Radius of casting=[tex]\frac{Diameter}{2}=4 in[/tex]

Depth of casting=4 in

We have to find the distance of light bulb should be from the vertex

It means the reflector passing through the point (4,4).

Equation of parabola along y-axis is given by

[tex]x^2=4ay[/tex]

Using the equation substitute x=4 and y=4

[tex]16=16a[/tex]

[tex]a=\frac{16}{16}=1[/tex]

The focus of parabola is at (0,a).

Therefore, the focus of reflector=(0,1)

Hence, the light bulb should be placed 1 in far  from the vertex.

The distance of light bulb can be calculated using equation of parabola. The parabola is a plane curve which is U-shaped.

The distance of light bulb is [tex]1\:\rm in[/tex].

Given:

The diameter is [tex]8\:\rm in[/tex].

The radius is [tex]=\frac{d}{2}=\frac{8}{2}=4 \:\rm in[/tex].

The depth is [tex]4\:\rm in[/tex].

Since the depth is [tex]4\:\rm in[/tex] and radius is [tex]4\:\rm in[/tex] so reflector passes through the point [tex](4,4)[/tex].

Write the equation of parabola for y-axis.

[tex]x^2=4ay[/tex]

Putting [tex]4[/tex] for [tex]x[/tex] and [tex]4[/tex] for [tex]y[/tex].

[tex]4^2=4\times a\times 4\\a=\frac{16}{16}\\a=1[/tex]

The focus of a parabola is [tex](0,1)[/tex].

The distance of light bulb is [tex]1\:\rm in[/tex].

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Answer:

[tex]m\angle DEC=91^o[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The measure of the interior angle is the semi-sum of the arcs comprising it and its opposite

so

[tex]m\angle DEC=\frac{1}{2}[arc\ AB+arc\ DC][/tex]

substitute the given values

[tex]m\angle DEC=\frac{1}{2}[39^o+143^o][/tex]

[tex]m\angle DEC=\frac{1}{2}[182^o][/tex]

[tex]m\angle DEC=91^o[/tex]

Answer:

[tex]\displaystyle \frac{12}{13} = sin\:θ\:[0,3743340836π ≈ θ][/tex]

Step-by-step explanation:

Remember this?!

Extended Information on Trigonometric Ratios

[tex]\displaystyle \frac{OPPOSITE}{HYPOTENUSE} = sin\:θ \\ \frac{ADJACENT}{HYPOTENUSE} = cos\:θ \\ \frac{OPPOSITE}{ADJACENT} = tan\:θ \\ \frac{HYPOTENUSE}{ADJACENT} = sec\:θ \\ \frac{HYPOTENUSE}{OPPOSITE} = csc\:θ \\ \frac{ADJACENT}{OPPOSITE} = cot\:θ[/tex]

I am joyous to assist you anytime.

answer : 8 vans and 7 buses

Answer:

  1)  B, C, and D

  2)  plane BFD

Step-by-step explanation:

1) Collinear points are points on the same line. There are two lines shown. One of them shows points A, C, and E on it. The other line shows points B, D, and D on it. The latter set of points is listed among the answer choices.

__

2) A plane can be named by three points in the plane that are not on the same line. The points shown as being in the plane are B, C, D, F, with points B, C, and D being on the same line (see question 1). So, the plane can be named with point F and any two of B, C, and D. Plane BFD is an appropriate name.

Answer:

-20% ?

Step-by-step explanation:

I'm not sure but 2% of 5 is 1 so 20% of 500 is 100. so minus 20% would give you 400.

To find the percent of change in foreclosures, subtract the new number from the old number, divide by the old number, and multiply by 100. The foreclosures went from 500 to 400, resulting in a 20 percent decrease.

To calculate the percent of change in the number of foreclosures from one year to the next, you take the difference in the numbers (new number - old number), divide by the old number (the number from the earlier year), and then multiply the result by 100 to obtain a percentage. In this case, there were 500 foreclosures last year and 400 this year. Therefore, the calculation is as follows:

(Old Number - New Number) / Old Number × 100 = Percent Change

(500 - 400) / 500 × 100 = 100 / 500 × 100

0.2 × 100 = 20%

There has been a 20 percent decrease in foreclosures from last year to this year.

Answer: Use all numbers to add.

2918+49=2967

Estimated.... 2918+49 ≈ 3050

Hope this helps you out.

Answer:use all the numbers

Step-by-step explanation:

2918+49 ≈ 3050

Answer:The number of yards of Chris's ribbon are neither purple nor green is 1.625 yards

Step-by-step explanation:

Chris has a total of 3 2/3 yards of ribbon. Converting 3 2/3 yards to improper fraction, it becomes 11/3 yards. Chris has 7/8 yards of purple ribbon. He also has 1 1/6 yards of green ribbon. Converting to improper fraction, it becomes 7/6 yards of green ribbon.

The total length of purple and green ribbon that Chris has would be 7/8 + 7/6 = (21+28)/24 = 49/24

The number of yards of Chris's ribbon are neither purple nor green is 11/3 - 49/24 = (88 - 49)/24

= 39/24 = 1.625 yards

Answer:

There is no finite maximum value of the radius of C.

Step-by-step explanation:

We have been given that in the rectangular coordinate system, points (4,0) and (–4,0) both lie on circle C. We are asked to find the maximum possible value of the radius of C.

If the center of the circle C would have been at origin (0,0), then the radius of the circle C would be 4 units.

Since there is no information given about center of circle, so radius can be infinitely long. Therefore, there is no finite maximum value of the radius of C.

Answer:

The equation of the line is y = (-1/2)x + 5

Step-by-step explanation:

[tex]m = \frac{y2 - y1}{x2 - x1 } [/tex]

First of all, have to find gradient using the formula above :

(2,4) & (14,-2)

m = (-2-4) / (14-2)

= -6 / 12

= -1/2

Second, using y = mx + b as b is a constant and is a y-intercept. Using any of these 2 coordinates to find the value of b with given gradient :

y = mx + b

Let y=4 & x=2

4 = (-1/2)(2) + b

b = 4 + 1

= 5

Lastly, put the value of gradient and y-intercept into the equation :

y = mx + b

Let m=-1/2 & b=5

y = (-1/2)x + 5

Answer:

a) equilibrium price = $48

b) shortage of 50 million phones

Step-by-step explanation:

Quarterly sales of a brand/quantity demanded (q) =

-p+126 million phones

a) if supply(q) = 9p - 354 million phones, at equilibrium quantity demanded = quantity supplied

-p + 126 = 9p - 354

-p - 9p = -354 - 126

-10p = -480

p = -480/-10

p = $48

The price at equilibrium = $48

b) actual wholesale price in the fourth quarter of 2004= $43

Quantity demanded = -p + 126

= -43 + 126

= 83 million phones

Quantity supplied = 9p - 354

= 9(43) - 354

= 387 - 354

= 33 million phones

Since quantity supplied is less than quantity demanded, there will be a shortage.

Shortage = 83 -33

= 50 million phones

The equilibrium price is determined to be $48 by equating the demand and supply. At an actual price of $43, there is a shortage of 50 million phones. This is found by comparing the quantities demanded and supplied at that price.

Equilibrium Price and Surplus/Shortage Estimation

We are given the demand and supply functions for cell phones, where the demand function is q = −p + 126 and the supply function is q = 9p − 354. To find the equilibrium price, we set the demand function equal to the supply function:

Solving for p:

Hence, the equilibrium price is $48.

For part (b), we need to estimate the shortage or surplus at an actual wholesale price of $43. We substitute p = 43 into both demand and supply functions:

Therefore, there is a shortage of 83 - 33 = 50 million phones at the price of $43.

Answer:

y=7  number of hours at grocery store

x=18 number of hours at baby- sitting

Step-by-step explanation:

According to the information provided.

x is number of hours at baby- sitting

y is number of hours at grocery store

total number of hours worked

1) x+y =25

total earn in a week

2) x*$6 + y* $9 = $171

from equation 1

x+y=25

x= 25-y

we place the above derived equation in equation 2

x*$6 + y* $9 = $171

(25-y)*$6 + y* $9 = $171

(25*6) -6y +9y =171

150+3y=171

3y=171-150

3y=21

y=7  number of hours at grocery store

x= 25-y

x= 25-7

x=18 number of hours at baby- sitting

Answer:

Length of the lake is 97.30 m

Step-by-step explanation:

We have given Caleb starts at one end of the lake and walk 95 m

So [tex]d_1=95m[/tex]

And then he turns at an angle of 60°

So [tex]\Theta =60^{\circ}[/tex] and then again walk 8 m

So [tex]d_2=8m[/tex]

We have to fond the total length of the lake , that is d

Total length of the lake is given by [tex]d=\sqrt{d_1^2+d_2^2+2d_1d_2cos\Theta }=\sqrt{95^2+8^2+2\times 95\times 8\times cos60^{\circ}}=97.30m[/tex]

So length of the lake is 97.30 m

Final answer:

Using the Law of Cosines with Caleb's path measurements of 95 meters and 103 meters at a 60-degree angle, the length of the lake is found to be approximately 104 meters.

Explanation:

To estimate the length of the lake, we can represent Caleb's path as a triangle, where the length of the lake forms one side of the triangle. Caleb starts by walking 95 meters along one side, then makes a 60° angle and walks 8 meters more than the length of the first path, forming the second side of the triangle. The length of the lake, which is the final side, can be calculated using the Law of Cosines.

The Law of Cosines is given by c² = a² + b² - 2ab×cos(γ), where γ is the enclosed angle and a, b, and c are the lengths of the sides of the triangle.

In this case, a = 95m, b = 95m + 8m = 103m, and γ = 60°. Plugging these values into the Law of Cosines, we will find the length of the lake (c).

c² = 95² + 103² - 2×95×103×cos(60°)

c² = 9025 + 10609 - 19570×0.5

c² = 9025 + 10609 - 9785

c² = 10849

c ≈ √10849

c ≈ 104.16 meters

Therefore, the length of the lake is approximately 104 meters long.