What is the range of this absolute value function?
A. -6 ≤ y < ∞
B. -∞ < y ≤ -6
C. 5 ≤ y < ∞
D. -∞ < y < ∞

Answer:

a)  see below

b)  radius = 16.4 in (1 d.p.)

c)  18°. Yes contents will remain. No, handle will not rest on the ground.

d)  Yes contents would spill.  Max height of handle = 32.8 in (1 d.p.)

Step-by-step explanation:

Part a

A chord is a line segment with endpoints on the circumference of the circle.  

The diameter is a chord that passes through the center of a circle.

Therefore, the spokes passing through the center of the wheel are congruent chords.

The spokes on the wheel represent the radii of the circle.  Spokes on a wheel are usually evenly spaced, therefore the congruent central angles are the angles formed when two spokes meet at the center of the wheel.

Part b

The tangent of a circle is always perpendicular to the radius.

The tangent to the wheel touches the wheel at point B on the diagram.  The radius is at a right angle to this tangent.  Therefore, we can model this as a right triangle and use the tan trigonometric ratio to calculate the radius of the wheel (see attached diagram 1).

[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]

where:

Given:

Substituting the given values into the tan trig ratio:

[tex]\implies \sf \tan(20^{\circ})=\dfrac{r}{45}[/tex]

[tex]\implies \sf r=45\tan(20^{\circ})[/tex]

[tex]\implies \sf r=16.37866054...[/tex]

Therefore, the radius is 16.4 in (1 d.p.).

Part c

The measure of an angle formed by a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.

If the measure of the arc AB was changed to 72°, then the other intercepted arc would be 180° - 72° = 108° (since AC is the diameter).

[tex]\implies \sf new\: angle=\dfrac{108^{\circ}-72^{\circ}}{2}=18^{\circ}[/tex]

As the handle of the cart needs to be no more than 20° with the ground for the contents not to spill out, the contents will remain in the handcart at an angle of 18°.

The handle will not rest of the ground (see attached diagram 2).

Part d

This can be modeled as a right triangle (see diagram 3), with:

Use the sin trig ratio to find the angle the handle makes with the horizontal:

[tex]\implies \sf \sin (\theta)=\dfrac{O}{H}[/tex]

[tex]\implies \sf \sin (\theta)=\dfrac{48-r}{48}[/tex]

[tex]\implies \sf \sin (\theta)=\dfrac{48-45\tan(20^{\circ})}{48}[/tex]

[tex]\implies \theta = 41.2^{\circ}\:\sf(1\:d.p.)[/tex]

As 41.2° > 20° the contents will spill out the back.

To find the maximum height of the handle from the ground before the contents start spilling out, find the height from center of the wheel (setting the angle to its maximum of 20°):

[tex]\implies \sin(20^{\circ})=\dfrac{h}{48}[/tex]

[tex]\implies h=48\sin(20^{\circ})[/tex]

Then add it to the radius:

[tex]\implies \sf max\:height=48\sin(20^{\circ})+45\tan(20^{\circ})=32.8\:in\:(1\:d.p.)[/tex]

(see diagram 4)

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

Circle Theorem vocabulary

Secant: a straight line that intersects a circle at two points.

Arc: the curve between two points on the circumference of a circle

Intercepted arc: the curve between the two points where two chords or line segments (that meet at one point on the other side of the circle) intercept the circumference of a circle.

Tangent: a straight line that touches a circle at only one point.

Answer

16/9

Step-by-step explanation:

I don't know if this is right or not and I'm sry if its wrong

but 16/9 divided by 4 = 9/4 

Answer:

C

Step-by-step explanation:

Exponential functions have 2 things in common

-They increase infinitely once they approach a certain point

-They don't decrease anymore once they approach a certain point

the number of possible call letters of a radio station (8 letters long) if the first letter must be a W or a K and no letter is repeated is 4, 845, 456, 0000 call letter

From the given information, we need to find the number which is

First letter have only two possibility.

But we know that there are a total of 26 letters in alphabet.

First letter must be W or K. So remaining 25 letters.

The Remaining 7 letters can be any alphabet (25 letters)

We also know that np letters are repeated.

So, the  second letters have 25 possibility and the third letter have 24 possibility and so one;

Thus, we have

= 2 × 25 × 24 × 23 × 22 × 21 × 20 × 19

Multiply through

= 4, 845, 456, 0000 call letter

Thus, the number of possible call letters of a radio station (8 letters long) if the first letter must be a W or a K and no letter is repeated is 4, 845, 456, 0000 call letter

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x = 4 would be the solution of the given problem.

I can't speak to the first part of this question, as I don't totally have context for what they're asking, but we can solve for x using one of the laws of logarithms, namely:

log.m + log.n = log.mn

Using this law, we can combine and rewrite our initial equation as

log(x^2- x) = 3

Remember that logarithms are simply another way of writing exponents. The logarithm  is just another way of writing the fact . Keeping that in mind, we can express our logarithm in terms of exponents as

log(x^2- x) = 3 --> x^2- 2x = 8

2³ = 8, so we can replace the left side of our equation with 8 to get

x^2 - 2x - 8 =0

(x-4)(x+2) = 0

Hence x = 4 0r -2. As x cannot be negative so it would be 4.

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The area of the square is 100 inches square

The perimeter of a square is calculated using the formula;

Perimeter = 4 a

Where

a is the side

It is also important to know that the area of a square is calculated using the formula;

Area = a²

Where a represents the side of the square

Let's find the side by substituting the value of the perimeter in the formula, we have

40 = 4a

a = 40/4

a = 10 inches

Then , let's find the area by substituting the value of the side

Area = 10 ²

Area = 10 × 10

Area = 100 inches square

Thus, the area of the square is 100 inches square

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Answer:

x = 3

Step-by-step explanation:

Here ,

4x + 9 =7x

--> 4x - 7x = -9

--> -3x = -9

--> ≠ x = -9/3

--> x = 3

The probability of it landing heads up at least three times is 1/4.

According to the statement

We have a given that the coin is tossed 4 times And we have to find the probability of it landing heads up at least three times.

So, we know that the

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1.

So, According to the statement

Coin is tossed 4 times and we have to probability of find heads three times it means n = 3.

To get 3 heads, means that one gets only one tail. This tail can be either the 1st coin, the 2nd coin, the 3rd, or the 4th coin.

Thus there are only 4 outcomes which have three heads.

The probability is 4/16 = 1/4.

So, The probability of it landing heads up at least three times is 1/4.

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The largest possible area of one of the circles would be 314. 2 cm²

It is important to note that the formula for finding the area of a circle is given as;

Area = [tex]\pi r^2[/tex]

From the information given, we have the sides of the square to have a value of 20cm

Also note that the measure of the sides of the square is the size of the diameter of the circle

But we need to find the radius

radius = diameter/2

radius = 20/ 2 = 10cm

Substitute value of radius into the formula for area

Area = 3. 142 × 10^2

Area = 3. 142 × 10 × 10

Area = 314. 2 cm²

The area of the circle is 314. 2cm^2

Thus, the largest possible area of one of the circles would be 314. 2 cm²

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Answer:

1083 seats

Step-by-step explanation:

this is an arithmetic sequence :

an = an-1 + c

in our case

a1 = 10 (10 seats in the first row)

a2 = a1 + 1 = 10 + 1 = 11

a3 = a2 + 1 = a1 + 1 + 1 = 10 + 1 + 1 = 12

an = an-1 + 1 = a1 + (n-1)×1 = a1 + n - 1 = 10 + n - 1 = 9 + n

a38 = a1 + 37 = 10 + 37 = 47 seats.

altogether this means the sum of the arithmetic sequence of a1 to a38.

the general sum of an arithmetic sequence from a1 to an is

n×(a1 + an)/2

in our case

38×(10 + 47)/2 = 19×57 = 1083 seats

Surface area = 400 unit square

Surface area = 1100 unit square

Surface area = 1100 unit square

Surface area = 2400 unit square

The first figure is a cuboid;

Surface area of a cuboid = 2lw+2lh+2hw

L = 9

W = 11

h = 4

Surface area = 2( 9 × 11) + 2(9 × 4) + 2( 4× 11)

Surface area = 2(99) + 2(36) + 2(44)

Surface area = 198 + 72 + 88

Surface area = 400 unit square

Surface area of a triangular prism = Area of top + area of base

Area of the top = 1/ 2 b × h

Area = 1/2 × 4 × 8

Area = 16 unit square

Area of base = length × width

Area = 4× 4

Area = 16 unit square

Surface area = 16 + 16

Surface area = 1100 unit square

Surface area of a cylinder = 2πrh + 2πr² = 2 ( 3. 412 × 7. 5 × 15) + 2 ( 3. 141× 7. 5 × 7.5)

Surface area = 706. 95 + 353. 48

Surface area = 1100 unit square

Surface area of a cone =πr(r+ √h² + r2)

Surface area = [tex]3. 142 * 15 ( 15 + \sqrt{10^2 +34^2 }[/tex]

Surface area = [tex]47. 13 * 50. 44[/tex]

Surface area = 2400 unit square

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Answer:

7

Step-by-step explanation:

given a polynomial divide by (x - a) then the remainder is f(a)

here f(x) is divided by (x - 5) , so remainder is

f(5) = 2(5)³ - 12(5)² + 11(5) + 2

     = 2(125) - 12(25) + 55 + 2

     = 250 - 300 + 57

     = - 50 + 57

    = 7

Answer:

see explanation

Step-by-step explanation:

the equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k ) are the coordinates of the vertex and a is a multiplier

here (h, k ) = (1, - 4 ) and a = 1 , then

y = (x - 1)² - 4 → b

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

given 3 sides of a triangle then an angle may be found using the cosine law. → b

if the 3 sides are a, b, c then

cosA = [tex]\frac{b^2+c^2-a^2}{2bc}[/tex] ← allowing ∠ A to be found

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

cosC = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BC}{AC}[/tex] = [tex]\frac{4}{5}[/tex] → a

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

since the triangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{AB}{DE}[/tex] = [tex]\frac{BC}{EF}[/tex] ( substitute values )

[tex]\frac{x}{6}[/tex] = [tex]\frac{10}{4}[/tex] ( cross- multiply )

4x = 6 × 10 = 60 ( divide both sides by 4 )

x = 15 → b

Answer:

Step-by-step explanation:

There are a few algebraic, geometric, and trig relations you are expected to remember. These come into play in this set of questions.

When solving any problem, the first step is to understand what is being asked. The second step is to identify the relevant information and relationships that can help you answer.

You are asked for the equation of a parabola with a given vertex. The vertex form equation will be useful. We can assume a scale factor ('a') of 1.

For vertex (h, k) = (1, -4) and a=1, the vertex form equation is ...

  y = a(x -h)² +k

  y = 1(x -1)² +(-4)

  y = (x -1)² -4

You are given 3 sides and want to find an angle. The useful relation in this case is the Cosine Law. (If you wanted to use the Sine Law, you would already need to know an angle.)

The mnemonic SOA CAH TOA reminds you that the cosine relation is ...

  Cos = Adjacent/Hypotenuse

The side adjacent to angle C is marked 4; the hypotenuse is marked 5. The desired ratio is ...

  cos(C) = 4/5

The measure x is also the measure of side AB. The similarity statement lists those letters as the first two. It also lists the letters DE as the first two. The other given side in ΔABC is BC, corresponding to side EF in the smaller triangle. Corresponding sides are proportional, so we have ...

  AB/DE = BC/EF

  x/6 = 10/4

We can find the value of x by multiplying this equation by 6:

  x = 6(10/4) = 60/4

  x = 15

Please note that BC is the shortest side in ΔABC. This means x > 10. There is only one such answer choice. (No math necessary.)

The graph that models the pH of this solution is graph A.

It should be noted that a graph is a diagram such as a series of one or more points, lines, line segments, curves, or areas which represents the variation of a variable in comparison with that of one or more other variables.

The pH is a measure of how acidic or basic water is. It should be noted that the range goes from 0 - 14, with 7 being neutral. The pHs of less than 7 indicate acidity, while a pH of greater than 7 indicates a base. The pH is really a measure of the relative amount of free hydrogen and hydroxyl ions that are in the water.

In this case, x represents the concentration of the hydrogen ions. The first graph illustrates this.

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Complete question:

The ph of a particular solution is given by pH=-log(x-2) where x represents the concentration of the hydrogen ions in the solution in moles per liter. Which graph models the ph of this solution?

The first table is filled correctly and can be used to determine how long it takes for Elias and Niko to polish the silver.

The first table is filled correctly and can be used to determine how long it takes for Elias and Niko to polish the silver.

The rate is the unit rate.

Now, since Elias polishes all of the silver himself in 40 minutes, his rate is 1/40.

Similarly, Niko’s rate would be 1/50.

Their time is the same because they are working at the same time, and as such we can write that as t. Then, the part of the silver polished would be the rate * time, or:

Thus;

Part polished by Elias: ¹/₄₀ * t

Part Polished by Niko: ¹/₅₀ * t

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Answer:

Below in bold.

Step-by-step explanation:

Opposite  angles of a quadrilateral i a circle are supplementary so

x = 180 - 112 = 68 degrees.

y  = 180 - 81 = 99 degrees.

Answer:

2:7

Step-by-step explanation:

If 2 of the puppies are brindle that leaves 7 that are not out of the nine making the ratio 2:7.

Solving a system of equations we can see:

First, let's define the variables:

We can write the system of equations:

S = (3/4)*M

M = K - $18

S + M + K = $286.40

First, we can rewrite the second equation to get:

K = M + $18

Now we can replace the first and second equations into the third one:

(3/4)*M + M + (M + $18) = $286.40

Now we can solve this for M:

M*(1 + 1 + 3/4) = $286.40 - $18

M = $268.40*(4/11) = $97.60

Now we can find the other two values:

K = M + $18 =  $97.60 + $18 = $115.60

S =  (3/4)*M =  (3/4)*$97.60 = $73.20

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Answer:240.5

Step-by-step explanation:

Angle C = 87°(calculated)

AB=C

AC=220=b

Sine laws c\sin87=220\66

c=240.5

The length of AB of the triangle is 13 units.

The side AB of the triangle can be found using cosine law,

Therefore,

c² = a² + b² - 2ab cos C

c² = 7² +  8² - 2 × 7 × 8 cos 120

c² = 49 + 64 - 112 cos 120

c²= 113 - (-56)

c² = 113 + 56

c² = 169

c = √169

c = 13 units

Therefore, the value of AB is 13 units

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A survey of 969 adult investors asked how often they tracked their portfolio. The probability that an adult investor tracks his or her portfolio daily is 0.238.

From the table it is clear that the number of responses daily by the adult investors is 231.

Probability is the ration of the number of possible value to the total value.

Here the number of daily response by the adult investors = 231

The total adult investors = 969

Hence, the The probability that an adult investor tracks his or her portfolio daily = 231/969 = 0.238.

Therefore, in the total number of 969 adult investors, the probability that the investors response daily is 0.238.

Disclaimer: The question is incomplete. The following is the correct question.

Question: Recently, the stock market took big swings up and down. A survey of 969 adult investors asked how often they tracked their portfolio. The table shows the investor responses. What is the probability that an adult investor tracks his or her portfolio daily

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Answer:

a ≈ 16.5 cm , b ≈ 23.8 cm

Step-by-step explanation:

using the Law of Sines in Δ ABC

[tex]\frac{a}{sinA}[/tex] = [tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex]

we require to calculate ∠ C

∠ C = 180° - (42 + 75)° = 180° - 117° = 63°

Then to find a

[tex]\frac{a}{sinA}[/tex] = [tex]\frac{c}{sinC}[/tex] ( substitute values )

[tex]\frac{a}{sin42}[/tex] = [tex]\frac{22}{sin63}[/tex] ( cross- multiply )

a × sin63° = 22 × sin42° ( divide both sides by sin63° )

a = [tex]\frac{22sin42}{sin63}[/tex] ≈ 16.5 cm ( to the nearest tenth )

similarly to find b

[tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex] ( substitute values )

[tex]\frac{b}{sin75}[/tex] = [tex]\frac{22}{sin63}[/tex] ( cross- multiply )

b × sin63° = 22 × sin75° ( divide both sides by sin63° )

b = [tex]\frac{22sin75}{sin63}[/tex] ≈ 23.8 cm ( to the nearest tenth )

Answer:

Step-by-step explanation:

    [tex]\sf \boxed{\bf\dfrac{a}{Sin \ A}=\dfrac{b}{Sin \ B}=\dfrac{c}{Sin \ C}}[/tex]

Side 'a' faces ∠A.

Side 'b' faces ∠B.

Side 'c' faces ∠C.

We have to find ∠C using angle sum property of triangle.

  ∠C + 75 + 42 = 180

         ∠C +117   = 180

                  ∠C  = 180 - 117

                 ∠C = 63°

[tex]\sf \dfrac{a}{Sin \ 42}= \dfrac{22}{Sin \ 63}\\\\ \dfrac{a}{0.67}=\dfrac{22}{0.89}\\\\[/tex]

   [tex]\sf a = \dfrac{22}{0.89}*0.67\\\\ \boxed{a = 16.56 \ cm }[/tex]

                        [tex]\sf \dfrac{b}{Sin \ B} = \dfrac{c}{Sin \ C}\\\\ \dfrac{b}{Sin \ 75}=\dfrac{22}{Sin 63}\\\\ \dfrac{b}{0.97} =\dfrac{22}{0.89}\\\\[/tex]

                             [tex]\sf b = \dfrac{22}{0.89}*0.97\\\\ \boxed{b =23.98 \ cm }[/tex]

Step-by-step explanation:

that means nothing else than

y = k×x

24 = k×6

k = 24/6 = 4

y = k×-4 = 4×-4 = -16

The missing length in the right triangle as given in the task content is; 156.

It follows from the complete question that the triangle given is a right triangle and the missing length (longest side) can be evaluated by means of the Pythagoras theorem as follows;

x² = 144² + 60²

x² = 20736 + 3600

x² = 24,336

x = √24336

x = 156.

Remarks: The complete question involves a right triangle and the missing length is the longest side.

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The value of the expression 8 (1/4 * 3/5) is  6/5

The expression is given as:

8 (1/4 * 3/5)

The associative property states that:

a * (b * c) = (a * b) * c

Using the above expression, we have:

8 (1/4 * 3/5) =  8 * 1/4 * 3/5

Evaluate the product

8 (1/4 * 3/5) =  2 * 3/5

This gives

8 (1/4 * 3/5) =  6/5

Hence, the value of the expression 8 (1/4 * 3/5) is  6/5

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Answer:

Trapezium, Rhombus, Kite, etc.

Step-by-step explanation:

A four - sided figure.

The name of a counterexample for the following biconditional: "A figure is a quadrilateral if and only if it is a polygon" is Triangle.

Used the concept of the polygon that states,

In geometry, a polygon can be defined as a flat or plane, two-dimensional closed shape bounded by straight sides.

Given the condition is,

"A figure is a quadrilateral if and only if it is a polygon"

Now, we know that;

If a polygon is a quadrilateral, then it has four sides, and if a polygon has four sides, then it is a quadrilateral.

Hence, Triangle is a counterexample for the biconditional "A figure is a quadrilateral if and only if it is a polygon."

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Answer:

[tex]\sf -3xy\left(2x^2-3xy+4y^2\right)[/tex]

Step-by-step explanation:

[tex]\sf -6x^3y+9x^2y^2-12xy^3[/tex]

To factor the GCF of 6x³y + 9x²y2 - 12xy³ let's apply the exponent rule:-

[tex]\boxed{\sf a^{b+c}=a^ba^c}[/tex]

[tex]\boxed{\sf x^3y=xx^2y,\:x^2y^2=xxyy,\:xy^3=xyy^2}[/tex]

[tex]\sf -6xx^2y+9xxyy-12xyy^2[/tex]

Rewrite,

[tex]\sf 2\cdot \:3xx^2y+3\cdot \:3xxyy+4\cdot \:3xyy^2[/tex]

Now, factor out the common term [tex]\sf -3xy[/tex]:-

__________________________

Answer:

Step-by-step explanation:

Trigonometry is a branch of mathematics that relates the measures of sides and angles in a triangle. The various trig functions of an angle are defined in terms of the sides of a right triangle containing that angle. Consequently, the Pythagorean theorem can be used to relate certain trig functions to each other, and to obtain formulas for the sum and difference of angles.

The Pythagorean theorem relates side measures in right triangles. For relating side measures to each other or to angles in non-right triangles, one or both of the Law of Sines and the Law of Cosines must be used.

  both b) and c)

The Law of Sines relates side lengths and the sines of angles. Knowing three side lengths tells you nothing about the angles except the ratio of their sines.

The Law of Cosines relates three side lengths and one angle. Knowing the three side lengths, the angle measure can be found.

There is no Tangent Law.

Trigonometry is a calculation tool, not a measuring tool. As such, it allows calculation of angles and sides (of a triangle) that cannot be physically measured.

Paintings benefit greatly from rules of proportion and perspective. In general, trigonometry is not directly involved. (Computer-generated art may implement these rules of perspective using trigonometry. Paintings, in general, make no direct use of trigonometry.)

The requried simplified expression for  [tex]y^{5/4}[/tex]  is given by [tex]y^{5/4}= 32 * 10^{10k}[/tex]

To find an expression for [tex]y^{5/4}[/tex] in terms of k, we'll substitute the given value of y into the expression and then apply the exponent (5/4).

Given: [tex]y = 16 *10^{8k}[/tex]

Now, let's calculate [tex]y^{5/4}[/tex]:

[tex]y^{5/4} = (16 *10^{8k})^{5/4}[/tex]

To apply the exponent (5/4), we raise each factor to the power of (5/4):

[tex]y^{5/4}= 16^{5/4} * (10^{8k})^{5/4}[/tex]

Since [tex](10^8)^k[/tex] is [tex]10^{(8k)}[/tex], we have:

[tex]y^{5/4}= 16^{5/4}* 10^{8k * 5/4)[/tex]

[tex]y = 16*10^{8k}[/tex]

[tex]y^{5/4}= 2^5 * 10^{10k}[/tex]

Finally, we can write the expression in terms of k:

[tex]y^{5/4}= 32 * 10^{10k}[/tex]

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