Peter and Myra are on their school's activity planning committee. They are polling each
homeroom class to find how many students are in after-school clubs. In Peter's class, 9 out of
the 20 students are in clubs. In Myra's class, 12 out of the 24 students are in clubs. Do the
two classes have the same ratio of students in clubs to total students?
yes
No

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]:-

__________________________

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

Answer:

8.75

Step-by-step explanation:

3.75/3=1.25

1.25=1 square feet

1.25(7)=8.75

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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Amie's error while measuring the volume of a sphere is that Amie should have multiplied 54 by 2/3. Thus, the first option is the right choice.

In the question, we are given that a sphere and a cylinder have the same radius and height.

We assume the radius of the sphere to be r, and its height to be h.

Now, the height of a sphere is its diameter, which is twice the radius.

Thus, the height of the sphere, h = 2r

Given that the sphere and the cylinder have the same radius and height, the radius of the cylinder is r, and its height is 2r.

The volume of a sphere is given by the formula, V = (4/3)πr³, where V is its volume, and r is its radius.

Thus, the volume of the given sphere using the formula is (4/3)πr³.

The volume of a cylinder is given by the formula, V = πr²h, where V is its volume, r is its radius, and h is its height.

Thus, the volume of the given cylinder using the formula is πr²(2r) = 2πr³.

Now, to compare the two volumes we take their ratios, as

Volume of the sphere/Volume of the cylinder

= {(4/3)πr³}/{2πr³}

= 2/3.

Thus, the volume of the sphere/the volume of the cylinder = 2/3,

or, the volume of the sphere = (2/3)*the volume of the cylinder.

Given the volume of the cylinder to be 54 m³, Amie should have multiplied 54 by 2/3 instead of adding the two.

Thus, Amie's error while measuring the volume of a sphere is that Amie should have multiplied 54 by 2/3. Thus, the first option is the right choice.

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For the complete question, refer to the attachment.

The value of 2+8+32+128…..,n=9 is 174762

The series is given as:

2+8+32+128…..,n=9

Start by calculating the common ratio (r)

r = 8/2

r = 4

The sum of the series is then calculated as:

[tex]S_n = \frac{a *(r^n - 1)}{r -1}[/tex]

This gives

[tex]S_9 = \frac{2 *(4^9 - 1)}{4 -1}[/tex]

Evaluate the difference

[tex]S_9 = \frac{2 *( 262143)}{3}[/tex]

Evaluate the quotient

[tex]S_9 = 174762[/tex]

Hence, the value of 2+8+32+128…..,n=9 is 174762

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Answer: △JKL & △ MNP

Step-by-step explanation:

△JKL is similar because:

5 x 1.6 = 8

2.5 x 1.6 = 4

3.75 x 1.6 = 6

△MNP is also similar because:

4 x 2 = 8

2 x 2 = 4

3 x 2 = 6

Reduce the numbers in the list modulo 3 to get the set

{1, 2, 0, 1, 2, 0, …, 1, 2, 0}

containing [tex]k[/tex] copies each of 1, 2, and 0.

Take any 3 elements from the list. Their sum is divisible by 3 if those elements' residues also sum to 3 ≡ 0 (mod 3). To get a sum of 0, we must make one of the following choices:

0 + 0 + 0 ≡ 0 (mod 3)

1 + 1 + 1 ≡ 3 ≡ 0 (mod 3)

2 + 2 + 2 ≡ 6 ≡ 0 (mod 3)

0 + 1 + 2 ≡ 3 ≡ 0 (mod 3)

There are

[tex]\dbinom k3 \dbinom k0 \dbinom k0 = \dfrac{k(k-1)(k-2)}6[/tex]

ways of choosing 3 elements with a given residue and 0 elements with any other residue, hence

[tex]3\dbinom k3\dbinom k0\dbinom k0 = \dfrac{k(k-1)(k-2)}2[/tex]

ways of choosing any 3 elements with the same residue, and there are

[tex]\dbinom k1 \dbinom k1 \dbinom k1 = k^3[/tex]

ways of choosing any 3 elements with distinct residues.

So, the total number of ways of making the selection is

[tex]3\dbinom k3\dbinom k0^2 + \dbinom k1^3 = \boxed{\dfrac32 k^3 - \dfrac32 k^2 - k}[/tex]

Answer:

8

Step-by-step explanation:

because hhhhhh

gghhhh

jkkkkk

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vvvvv

dddfff

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kkkkkk

lllll.ssss

A population of a particular yeast cell develops with a constant relative growth rate of [tex]0.4311[/tex] per hour. The initial population consists of [tex]3.9[/tex]million cells. Find the population size (in millions of cells) after 5 hours. Population size after [tex]5hrs[/tex]  is  [tex]33.6million[/tex].

How can we find the population size ?

The projection of population growth in yeast is given by

[tex]N=N_{0} e^{rt}[/tex]

Where [tex]N_{0}[/tex]=initial population which is [tex]3.9million[/tex]

[tex]r[/tex]=intrinsic rate of natural increases which is [tex]0.4311million per hour[/tex]

N is population size

Substitute the values

[tex]N=N_{0} e^{rt}\\N=3.9(e^{0.4311*5} )\\\\N=33.6 million[/tex]

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1a) f(x) = -1/3(x + 12)² + 9

1b) f(x) = -1/3x² - 8x - 39

Lets simplify it,

Expand by FOIL (First Outside Inside Last)

Standard Form: ax² + bx + c = 0

Transformations Graph: f(x) = a(bx - c)² + d

Reflected down and vertically stretched by 1/3:   a = -1/3

Shifted vertically by 9 units:   d = 9

Shifted horizontally by -12 units:   c = -12

Vertex Form:

 f(x) = a(bx - c)² + d

 f(x) = -1/3(x + 12)² + 9

Standard Form:

f(x) = -1/3(x + 12)² + 9

f(x) = -1/3(x² + 24x + 144) + 9

f(x) = -1/3x² - 8x - 48 + 9

f(x) = -1/3x² - 8x - 39

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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:

22 millilitres of 13% solution

23 millilitres of 19% solution

Step-by-step explanation:

A + B = 45

13A + 19B = 45 x 17 = 765

-13A - 13B + 13A + 19B = -650 + 765 = 115

5B = 115

B = 23

22 + 23 = 45

A = 22

Answer:

a= 100.2

Step-by-step explanation:

You solve it like you ate looking for the hypotenuse

Answer: 9 triangles.

Step-by-step explanation:

Just make a dot in the middle and make lines and connect each line to an edge or bend.

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 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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The equivalent form of the equation y=[tex]x^{2} -2x-15[/tex]given is y=(x+3)(x-5).

Given an equation y=[tex]x^{2}[/tex]-2x-15 and we are required to find the equivalent form of the equation.

Equation is like a relationship between two or more variables that are expressed in equal to form. Equation of two variables look like ax+by=c. It may be linear equation, quadratic equation, cubic equation or any other equation  depending on the powers of the variable.

To find the equivalent equations we are required to form factors of the equation. Equivalent equation are those equations which when solved gives the same solution as the equation when solved gives.

y=[tex]x^{2}[/tex]-2x-15

y=[tex]x^{2}[/tex]-5x+3x-15

y=x(x-5)+3(x-5)

y=(x+3)(x-5)

Hence the equivalent form of the equation y= [tex]x^{2}[/tex]-2x-15 given is

y=(x+3)(x-5).

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Question is incomplete as question should includes the equation

y= [tex]x^{2}[/tex]-2x-15.

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]

The value of the Mean & Standard deviation is 1.30 and 1.18.

According to the statement

we have a given that the random variable x represents the number of computers that families have along with the corresponding probabilities.

And we have to find the mean and standard deviation from the given data which is related to the probabilities values

So, according to the given data

The formula to compute the mean is:

Mean = summation [x*p(x)]

Compute the mean as follows:

Mean = summation [x*p(x)]

Mean = summation [0*0.49 + 1* 0.05 + 2*0.32 + 3*0.07 + 4*0.07]

Mean = 0 +0.05 + 0.64 + 0.21 + 0.28

Mean = 1.18

The mean of the random variable x is 1.18.

And after calculating the variance from the formula get

The value of standard deviation is 1.30

So, The value of the Mean & Standard deviation is 1.30 and 1.18.

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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

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?

Answer:

(-2, 2)

Step-by-step explanation:

Given system of equations:

[tex]\begin{cases}y=x+4\\y=-2x-2 \end{cases}[/tex]

To solve by substitution, substitute the first equation into the second equation:

[tex]\implies x+4=-2x-2[/tex]

Add 2x to both sides:

[tex]\implies x+4+2x=-2x-2+2x[/tex]

[tex]\implies 3x+4=-2[/tex]

Subtract 4 from both sides:

[tex]\implies 3x+4-4=-2-4[/tex]

[tex]\implies 3x=-6[/tex]

Divide both sides by 3:

[tex]\implies \dfrac{3x}{3}=\dfrac{-6}{3}[/tex]

[tex]\implies x=-2[/tex]

Substitute the found value of x into the first equation and solve for y:

[tex]\implies y=-2+4[/tex]

[tex]\implies y=2[/tex]

Therefore, the solution to the given system of equations is (-2, 2).

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Substitute y value from first eqn in second equation

Put in first one

(-2,2) is the solution

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: 6 bees

Step-by-step explanation:

The picture has 12 bees and 10 flowers. Thus, the ratio of flowers to bees is 10 to 12, which simplifies to 5 to 6.

Therefore, for every 5 flowers, there are 6 bees.

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:

Step-by-step explanation:

The sum of an alternating geometric series SUM((-1)^n*ar^n) = a/(1+r). The given series has r=2/3 and a=1. The sum will be 1/(1+2/3)= 3/5

Hello,

We have s0 = 1 and q = -2/3

[tex]S _{n} = S _{0} \times q {}^{n} = 1 \times ( - \frac{2}{3} ) {}^{n} [/tex]

[tex]S _{8} = ( - \frac{2}{3} ) {}^{8} = \frac{2 {}^{8} }{3 {}^{8} } = \frac{256}{6 561} [/tex]

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.)

Answer:

5x - y = 9

Step-by-step explanation:

the equation of a line in standard form is

Ax + By = C ( A is a positive integer and B, C are integers )

given

y = 5x - 9 ( subtract 5x from both sides )

- 5x + y = - 9 ( multiply through by - 1 )

5x - y = 9 ← in standard form