find a cartesian equation for the curve and identify it. r = 7 cos(θ)

The given curve is a second order, so it is a parabola. The area of the largest rectangle is 32 square units.

What is a curve?

A curve is a mathematical object that is similar to a line but does not have to be straight. A curve can be thought of intuitively as the trace left by a moving point.

Given curve is y = 12 - x².

The vertex of the parabola is on the y-axis. Thus y-axis is a symmetric line of the parabola.

Thus the x-coordinate of the vertices that are lying on the x-axis is the same but the opposite sign.

Thus the y-coordinate of the vertices that are lying on the x-axis is the same.

Thus the x-coordinate of the vertices that are lying on the curve is the same but the opposite sign.

Thus the y-coordinate of the vertices that are lying on the curve is the same.

The y - coordinate of the vertices that are lying on the curve is in form 12 - x².

Assume that the coordinate of the rectangle are A(a,0), B(-a,0), C(-a, 12 - a²), and D(a,12-a²).

The length of the rectangle is 2a.

The width of the rectangle is 12-a².

The area of the rectangle is A = 2a(12-a²).

A = 24a - 2a³

Differentiate with respect to a:

dA/da = 24 -6a² ....(i)

Equate dA/da with zero to find the critical point:

24 -6a² =0

6a² =24

a² = 4

a = ±2

Putting a=2 in A = 24a - 2a³:

A = 24×2 - 2×2³

A = 32

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The given curve is a second order, so it is a parabola. The area of the largest rectangle is 32 square units.

A curve is a mathematical object that is similar to a line but does not have to be straight. A curve can be thought of intuitively as the trace left by a moving point.

Given curve is y = 12 - x²

The vertex of the parabola is on the y-axis. Thus y-axis is a symmetric line of the parabola.

Thus the x-coordinate of the vertices that are lying on the x-axis is the same but the opposite sign.

Thus the y-coordinate of the vertices that are lying on the x-axis is the same.

Thus the x-coordinate of the vertices that are lying on the curve is the same but the opposite sign.

Thus the y-coordinate of the vertices that are lying on the curve is the same.

The y - coordinate of the vertices that are lying on the curve is in form 12 - x².

Assume that the coordinate of the rectangle are A(a,0), B(-a,0), C(-a, 12 - a²), and D(a,12-a²).

The length of the rectangle is 2a.

The width of the rectangle is 12-a².

The area of the rectangle is A = 2a(12-a²).

A = 24a - 2a³

Differentiate with respect to a:

dA/da = 24 -6a² ....(i)

Equate dA/da with zero to find the critical point:

24 -6a² =0

6a² =24

a² = 4

a = ±2

Putting a=2 in A = 24a - 2a³:

A = 24×2 - 2×2³

A = 32

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Complete question is: find the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y=12−x

2

A collection of six objects can be chosen from one of 120 possibilities. These six configurations comprise BDF, BFD, DBF, DFB, FBD, and FDB.

A permutation is what? The number of possible arrangements for a given set is calculated mathematically, and this process is known as permutation. Simply said, a mutation is a term that refers to the variety of possible arrangements or orders. The arrangement's order is important when using permutations.

Out of the six things A, B, C, D, E, and F, rotations of items B, D, and F (a total of three items) must be listed.

Simply said, we must find every configuration for these three elements.

The three objects can be placed in three ways, which equals (3x2x1) ways, or six.

These six configurations are BDF, BFD, DBF, DFB, FBD, and FDB.

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The linear function that passes through (9, 12) and is perpendicular to y=4  is given as follows:

x = 9.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

For which the parameters are given as follows:

The line y = 4 is an horizontal line, hence a perpendicular line will be a vertical line, which has the format given as follows:

x = c.

The point is (9,12), meaning that the x-coordinate is of 9, and thus the constant c is given as follows:

c = 9.

Hence the equation is:

x = 9.

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1/12 OR 0.0083% PROBABILITY is that  sit down in correct age order either clockwise.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true.

An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

The likelihood that an event will occur increases with its probability. A straightforward illustration is tossing a fair (impartial) coin.

The coin is fair, thus the outcomes "heads" and "tails" are equally likely; the likelihood of "heads" is equal to the likelihood of "tails"; and because there are no other conceivable outcomes, the likelihood of either "heads" or "tails" is 1/2 (which is also an acceptable spelling).

According to our question-

1/5 * 1/4 * 1/3 * 1/2 * 1/1 = 1/120

0.0083

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To simplify the expression 3k + 9 - 2k + 11, we need to combine like terms and simplify the expression.

The first step is to combine the constants (numbers without variables)

3k + 9 - 2k + 11 = 3k - 2k + 9 + 11

The next step is to combine the variable k

3k - 2k = k

Now we have

k + 9 + 11 = k + 20

The final simplified expression is k+20

Answer:

20

Step-by-step explanation:

3k+(-2k) = 1k or just k

+9+11 = +20

= k+20

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Given triangles, ΔDEF and ΔABC are Similar triangles.

A set of triangles that are the same in shape but different in size is known as the Similar Triangles. In other words, triangles that resemble one another but may not be exactly the same size are said to be comparable triangles. Congruent triangles and similar triangles generally differ from one another. When two triangles are similar triangles then the corresponding angles are equal.

Here we have

ΔDEF and ΔABC  

From the given figures

=> ∠D = ∠A

=> ∠E = ∠B

=> ∠F = ∠C  

Therefore,

Given triangles, ΔDEF and ΔABC are Similar triangles.

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This question will require divergent thinking.

The other questions are convergent thinking questions because they require a single, correct answer. However, the question "What are the various uses of a coffee mug?" requires divergent thinking because there is no single, correct answer. Instead, this question requires the respondent to think creatively and come up with a variety of possible uses for a coffee mug.

The question “What are the various uses of a coffee mug?” requires divergent thinking as it requires creative thinking and the generation of a variety of possible uses for a coffee mug.

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As per the unitary method the number of students who attended the play is 280.

The unitary method is a mathematical method used to find the value of a single unit in a given set of values. In this case, we will be using it to find the number of student tickets sold at the local play.

Let's say that "x" represents the number of student tickets sold.

The cost of each student ticket is $6, so the total cost of student tickets is $6x. The total cost of adult tickets is $11y, where "y" represents the number of adult tickets sold. The total ticket sales for the play was $3,000, so we can write the equation:

$6x + $11y = $3,000

We already know the cost of student tickets and adult tickets, so we can substitute those values into the equation:

$6x + $11y = $3,000

$6x = $3,000 - $11y

Now we can divide both sides of the equation by $6 to find the value of x:

x = ($3,000 - $11y) / $6

Since the total ticket sales were $3,000 for 400 tickets, we can use that information to solve for y:

$3,000 = $6x + $11y

$3,000 = $6x + $11(400 - x)

Expanding the equation:

$3,000 = $6x + $4400 - $11x

Combining like terms:

$3,000 = $-5x + $4400

Adding $5x to both sides:

$3,000 + $5x = $4400

Subtracting $3,000 from both sides:

$5x = $4400 - $3,000

$5x = $1400

Dividing both sides by 5:

x = $1400 / $5

x = 280

So, 280 students attended the play.

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If we fix x = 0 and z = 0 only y coordinate can change. So triple (0, y, 0) can only represent a point in y axis.

What do you mean by a plane?

In geometry, a plane is a surface made up of all the lines that are parallel to one another and connect any two locations on it. It is, in other words, a level or flat surface.

                          A plane is defined uniquely through any of the following in a Euclidean space of any number of dimensions: with the aid of three non-collinear points.

For the triple  (x,y,z ) to represent the point on the y - axis .

    x must be 0    y must be a real number

  and z must b 0

For the triple  (x,y,z ) to represent the point on the y - axis .

    x must be 0    y must be 0

  and z must be  a real number

For the triple  (x,y,z ) to represent the point on the y - axis .

    x must be real number     y must be 0

  and z must be  a real number

For the triple  (x,y,z ) to represent the point on the y - axis .

    x must be 0     y must be real number

  and z must be  a real number

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The particle's total d2 distance for the course of the time period indicated above, in accordance with the supplied assertion, was -3/2 m.

An example of a utility rule is one that generates a single output from a single input. The picture was obtained from Alex Federspiel. The equation y=x2 serves as an example of this. There is only one output for y for every input for x. We may assert that y appears to be a time variable given that x remain the source value. A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.

dx(t)/dt = 5t-8

[tex]\int_0^{x(t)} \mathrm{dx}(t)=\int_0^3 5 t-8 \mathrm{~d} t[/tex]

[tex]x(t)=\left.\left[\frac{5}{2} t^2-8 t\right]\right|_0 ^3[/tex]

f(0) = 0

[tex]f(3)=\frac{5}{2} * 3^2-8 * 3=-\frac{3}{2} m[/tex]

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The complete question is-

The velocity function v(t) (in meters per second) is given for a particle moving along a line.

V(t)= 5t-8, 0 <equal to 't' <equal to 3

Find the Total distance d2 traveled by the particle during the time interval given above?

I need help solving this problem, i know how to solve it but i keep getting the wrong answer. My answer is 457/10.. how do you solve this problem

Step-by-step explanation:

S = age of Spot

J(S) = the age of John, when Spot is S years old.

J(S) = S + 5

J(0) when Spot was born = 0 + 5 = 5

J(4) when Spot has 4th birthday = 4 + 5 = 9

John will be 9 years old.

Answer:

Step-by-step explanation:There are a total of 20 sweets in the bag (5 red + 6 blue + 9 yellow).

To find the probability that Sarah takes two sweets of the same color, we need to find the probability of her choosing a sweet of a specific color on her first pick and then the probability of her choosing another sweet of that same color on her second pick.

For example, if she chooses a red sweet on her first pick, the probability of that happening is 5/20, or 1/4. Then the probability of her choosing another red sweet on her second pick, given that she already picked one, is 4/20, or 2/10. Therefore the probability of her choosing two red sweets in a row is (1/4) * (2/10) = 1/20.

Similarly, the probability of her choosing two blue sweets in a row is (6/20) * (5/20) = 3/50.

The probability of her choosing two yellow sweets in a row is (9/20) * (8/20) = 9/100.

The probability of her choosing two sweets of the same color is the sum of the probability of her choosing two red sweets, two blue sweets, or two yellow sweets in a row.

P(Two same color) = 1/20 + 3/50 + 9/100 = 11/50

So, the probability of her choosing two sweets of the same color is 11/50.

Answer:

a. y = 2x - 6

To find the slope of the equation, we need to find the coefficient of x which is 2. So the slope of this equation is m=2

To find the y-intercept, we can substitute x = 0 into the equation:

y = 2(0) - 6

y = -6

So the y-intercept of this equation is b = -6

b. y = 1/2 * x + 12

To find the slope of the equation, we need to find the coefficient of x which is 1/2. So the slope of this equation is m=1/2

To find the y-intercept, we can substitute x = 0 into the equation:

y = 1/2(0) + 12

y = 12

So the y-intercept of this equation is b = 12

Therefore, the slope-intercept form of the equation is y = 2x - 6 and y = 1/2x + 12 respectively.

2.

D

Answer:

4

Step-by-step explanation:

Volume of the sphere to the nearest cubic cm will be 803.84 cm^3

A sphere has a spherical, symmetrical form. It is a three-dimensional solid with equal distances between each surface point and the center. On the basis of its radius, it has a surface area and a volume. There are no faces, corners, or edges on it.

The collection of all points in three-dimensional Euclidean space that are separated by a certain amount of space, or "radius," is known as a sphere (the "center"). The diameter is equal to twice the radius, and antipodes are pairs of locations on the sphere that are on opposing sides of a diameter.

Volume of sphere[tex]=\frac{4}{3} \pi r^3[/tex]

As we have the value of radius = r=4 cm

We have the value of total scoops = 3 mm

So, by putting the value in the above expression of the volume of the sphere, we will get it as:

[tex]V =\frac{4}{3} \times 3.14(4)^3 \times 3 \\=3.14 \times 256 \\=803.84 \mathrm{~cm}^3[/tex]

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The correct question should be:

Let's Scream for Ice Cream serves 3 scoops of ice cream in its signature cone. Each scoop is a sphere with a radius of 4 centimeters. To the nearest cubic centimeter, what is the total volume of ice cream served per cone?

A 1-U, 1-V graph is a special type of graph that has only one vertex of degree 1 (U) and one vertex of degree 2 (V). The edge connecting the two vertices is called the "bridge" of the graph.

Because the graph has only one vertex of degree 2, it forms a simple path (a straight line) between the two vertices. The vertex of degree 1 is a leaf, which means it is only connected to one vertex, and the vertex of degree 2 is the other end of the path. The path is a simple one, meaning it has no loops or multiple edges between the two vertices.

In a 1-U, 1-V graph, the vertex of degree 1 (U) can be thought of as the start of the path and the vertex of degree 2 (V) as the end of the path. Because there are no other edges or vertices in the graph, the path is the only structure in the graph and it's a straight line.

It's important to note that a 1-U, 1-V graph is a special case of a graph and it's not a general representation of all possible graphs. Most graphs have more than one vertex and multiple edges, making them more complex than a simple straight line.

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The most descriptive name would be a rhombus.

What is a rhombus?

A rhombus is a geometric shape that is a type of parallelogram. It is defined as a quadrilateral with all four sides congruent (equal in length). The opposite angles of a rhombus are also congruent. A rhombus can also be thought of as a square that has been stretched or compressed in one direction, so all its angles are right angles.

The most descriptive name for a quadrilateral with all congruent sides and two sets of parallel sides is a parallelogram. Specifically, a parallelogram with all congruent sides is called a rhombus.

Hence, The most descriptive name would be a rhombus.

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The volume of the given shape would be = 410cm³

A rectangle is defined as the type of quadrilateral that has two pairs of equal sides that are parallel to each other with angles equal to 90°.

From the shapes given above,

it is made up of three rectangles combine together with different lengths, width and height.

The volume of a rectangle = l×w×h

Rectangle A = 10×3×8 = 240

Rectangle B = 6×5×3 = 90

Rectangle C = 2×5×8 = 80

Therefore the volume of the shape = 240+90+80= 410cm³.

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The relative location of the drone are as follows;

Part A: The height of the drone will be 1.2 miles.

Part B: To be opposite distance from the top of the waterfall as the swimming site, the height of the drone should be 1.5 miles above the top of the water fall

Part C: The value 0 represents the location of the top of the water fall from which all other points are measured

Part D: Using the absolute function indicates that the distance to the site for swimming -1.5 miles is the same as the distance back, to the top which is 1.5 miles as |-1.5| = |1.5| = 1.5.

What is number line?

A number line is a diagram of a graduated straight line used to represent real numbers in elementary mathematics. It is assumed that every point on a number line corresponds to a real number, and that every real number corresponds to a point.

The height to which the drone needs to fly up in the air to be the opposite distance from the place for eating lunch is found has follows;

The distance from the location for eating lunch to the top of the water fall = 1.2 miles

The height to which the drone has to fly to be opposite from from where they eat lunch is 1.2 miles above the top of the waterfall

Part B: The height of the drone above the waterfall such that it is the same as the swimming site the to the top of the waterfall is 1.5 miles.

Part C; A zero is a reference point. The reference point in the question is location of the top of the waterfall, from which the distances of other location are sated

Part D: The distance from the water fall to the swimming site = -1.5 miles

Distance from the swimming site back = 1.5 miles

Using the absolute value function. we have;

The distance to the swimming site = |-1.5| = 1.5

The distance back up = |1.5| = 1.5

The absolute value function indicates that the distance in both direction is the same.

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Answer:

Equal

Step-by-step explanation:

Since all angles are the same

So the border lines should also be equal

The values of x and y are 1.8 units and 1.2 units respectively.

A perpendicular bisector is a line that bisects a line segment in two equal parts and makes an angle of 90° at the point of intersection. In other words, we can say that a perpendicular bisector divides a line segment at its midpoint making an angle of 90°.

Given that, RT = 15x + 5y, RS = 20x - 5y, WT = 40 and ST=21.

Here, RS=RT+ST

20x-5y=15x+5y+21

5x-10y=21 -------(I)

RT=ST

15x+5y=21 -------(II)

Multiply equation (II) by 2, we get

30x+10y=42 -------(III)

Add equation (I) and (III), we get

5x-10y+30x+10y=21+42

35x=63

x=63/35

x=9/5

x=1.8 units

Substitute x=1.8 in equation (I), we get

5(1.8)-10y=21

y=1.2 units

Therefore, the values of x and y are 1.8 units and 1.2 units respectively.

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Answer:

r=-7

Step-by-step explanation:

Let (x1,y1) be (1,r)and (x2,y2) be (5,1)

Slope m=2

[tex]slope = \frac{rise}{run} = \frac{y2 - y1}{ x2 - x1 } \\ = \frac{1 - r}{5 - 1} = 2 \\ 1 - r = 8 \\ r = - 7[/tex]

By the concepts of trigonometric ratios, the value of x in the right angle triangle is 1.

We know a right-angled triangle has three sides they are -: Hypotenuse,

Opposite and Adjacent.

We can remember SOH CAH TOA which is,

sin = opposite/hypotenuse, cos = adjecen/hypotenuse and

tan = opposite/adjacent.

If we take the reference angle of 60°, Then the hypotenuse is x and the adjacent is 2.

We know, cos = adjacent/hypotenuse.

Therefore,

cos60° = 2/x.

x = 2cos60°.

x = 2×1/2.

x = 1.

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Answer:

Step-by-step explanation:

The idea is, for each triangle, sort all of its sides and then store it in a map, if all those three sides are already present in the map then increase the frequency by 1, else its frequency will be 1. The count of elements of the map which have frequency 1 in the end will be the answer.

The fraction of blue marble in the jar is [(1/2)x]/ (x+3) where x+3 is the total marbles.

This question involves basic arithmetic and fractions.

We know that a fraction means a part of a whole. It is of the form a/b where b is not equal to zero and a, b are numbers.

Now, here let the total number of marbles in the bar is x .

Then , the number of blue marbles is (1/2)x i.e. 1/2 of x.

Now, if three black marbles are added to the jar, the total number of marbles = x + 3.

So, fraction of blue marble in the jar = (no. of blue marble)/(no. of total members)

= [(1/2)x]/ (x+3).

Thus, the fraction of blue marble in the jar is [(1/2)x]/ (x+3) where x+3 is the total marbles.

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The width of the tabletop is t feet when its length is 12 feet.

In contrast to width, which is used to describe how wide an object is, length simply states how long something is.

The term "width" is also used to refer to "breadth." It is the distance measured across a specific shape or object, such as a rectangle, whose lengths form right angles with the sides.

In contrast to the measurement of the area of the rectangle, which is a product of two units, the length and width. Length and width are two fundamental one-dimensional units. Today, a variety of length units are in use. The metre in SI units of measurement is the most fundamental of all.

Given that:

The length of a larger tabletop is 2 more than 2 times its width

l = 2(w) + 2

The length is 12

12 = 2(w) + 2

12 - 2 = 2w

10 = 2w

w = 10/2

w = 5

Thus, the width of the tabletop is t feet when its length is 12 feet.

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Full question:

The width of a larger tabletop is 2 more than 2 times its length. The length of the larger tabletop is 12 feet. What is its width, x?

The measure of angle BAC from the given rectangle ABCD is 39°.

A rectangle is a closed two-dimensional figure with four sides. The opposite sides of a rectangle are equal and parallel to each other and all the angles of a rectangle are equal to 90°.

In the given rectangle ABCD, diagonals AC and BD bisect each other E.

27) m∠DAC=2x+4 and m∠BAC=3x+1,

Here, m∠DAC+m∠BAC=90°

2x+4+3x+1=90°

5x+5=90°

5x=85

x=17

So, m∠BAC=3x+1=52

28) m∠BDC=7x+1 and m∠ADB=9x-7,

Here, m∠BDC+m∠ADB=90°

7x+1+9x-7=90

16x-6=90

16x=96

x=6

So, m∠BDC=7x+1=43°

29) m∠ABD=7x-31 and m∠CDB=4x+5,

Here, 7x-31=4x+5

3x=36

x=12

m∠ABD=7x-31 =53°

30) m∠BAC=x+3 and m∠CAD=x+15

Here, m∠BAC+m∠CAD=90°

x+3+x+15=90

2x+18=90

2x=72

x=36

m∠BAC=x+3=39°

Therefore, the measure of angle BAC from the given rectangle ABCD is 39°.

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The probability that it will snow on 3 consecutive days is 2.7 %

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty.

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen P(E)

= Number of favorable outcomes/Total Number of outcomes

Given, probability of 30% rain that it will snow on any day,

The probability of snow over a 3-day period is the product of the probabilities of rain for each day:

   P3-day = (P1-day)(P1-day)(P1-day)

                = (30/100)³

                = (3/10)³

                = 27/1000

                = 2.7%

Hence, 2.7% is the probability that it will snow on 3 consecutive days.

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The area of inscribed regular hexagon is 6[tex]\sqrt{3\\[/tex] unit square using the properties of equilateral triangles which combined, form the hexagon.

In a circle, a regular hexagon can be inscribed with all the six vertices of hexagon touching the circumference of the circle. The center of the circle is the center of the hexagon itself.

Now, if we join the center with all the vertices of the hexagon, we see six equilateral triangles are formed. The other two sides of each triangle apart from the ones joining the vertices of hexagon, are radii of the circle and all the sides are equal in length. The attached image can be referred for better understanding.

To find the area of the hexagon, we need to find the area of single equilateral triangle and multiply it by 6, as six triangles make a regular hexagon.

Radius of the circle ([tex]r[/tex]) = 2 units

Area of a single equilateral triangle = [tex]\sqrt{3} /4[/tex] [tex]r^{2}[/tex]

                                                           =( [tex]\sqrt{3} /4[/tex] ) × [tex]2^{2}[/tex]

                                                           = [tex]\sqrt{3}[/tex]

Area of hexagon = Area of 6 equilateral triangles = 6 × [tex]\sqrt{3}[/tex]

Area of hexagon = [tex]6\sqrt{3}[/tex] [tex]units^{2}[/tex]

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