You have $50 in your bank account.
You make $8 per hour mowing lawns.
How many hours must you mow lawns to have a total of $130 in your account?

Use the drop-down menus to complete the sentences below.

ANSWER

[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]

EXPLANATION

Given the points, y(-2,5),z(1,3)

[tex] ^{ \to} _{YZ} = \binom{1}{3} - \binom{ - 2}{5} = \binom{3}{ - 2} [/tex]

Therefore the ordered pair is <3,-2>

The magnitude is

[tex] |^{ \to} _{YZ}| = \sqrt{ {3}^{2} + ( - 2)^{2} } [/tex]

[tex] |^{ \to} _{YZ}| = \sqrt{ 9 +4} [/tex]

[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]

Answer: AAAAAAAAAAAAAAAAAAAAAAAAAAa

The area of shaded regions can be found using geometric principles or methods of integration depending on the actual shape and context. In most cases, area is proportional to the square of the distances. Integration techniques would be used if the shaded region is under a curve on a graph.

To find the area of the shaded regions, depending upon the shape and complexity of the region, you'd typically use geometric principles and calculations, potentially including those related to rectangles, triangles, circles, and/or other shapes. In some cases, these calculations might include figuring out the area of a larger shape and then subtracting the area of a smaller, non-shaded shape. For example, the area of a disc could be found by using the equation А = лr², and placing limits of integration from r = 0 to r = R in case the shaded area is comprised of thin rings of different radii. In other cases, you might be using principles of integration if the shaded region is under a curve on a graph, integrating the function f(x) from a certain lower limit x₁ to upper limit x₂. Also, keep in mind that the area is usually proportional to the square of the distances in a certain set-up.

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The domain of all three functions is all real numbers. The range of f(x) and h(x) is all real numbers, but the range of g(x) is all real numbers except 0.

The statements that accurately compare the domain and range of the functions are:

For the functions f(x) = 3x^2, g(x) = 1/3x, and h(x) = 3x:

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Answer:c and d

Step-by-step explanation:

i got it right

1 Simplify \frac{14}{30}

​30

​

​14

​​  to \frac{7}{15}

​15

​

​7

​​ .

\frac{7}{15}\div 14.00

​15

​

​7

​​ ÷14.00

2 Use this rule: a\div \frac{b}{c}=a\times \frac{c}{b}a÷

​c

​

​b

​​ =a×

​b

​

​c

​​ .

\frac{7}{15}\times \frac{1}{14.00}

​15

​

​7

​​ ×

​14.00

​

​1

​​  

3 Use this rule: \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}

​b

​

​a

​​ ×

​d

​

​c

​​ =

​bd

​

​ac

​​ .

\frac{7\times 1}{15\times 14.00}

​15×14.00

​

​7×1

​​  

4 Simplify 7\times 17×1 to 77.

\frac{7}{15\times 14.00}

​15×14.00

​

​7

​​  

5 Simplify 15\times 14.0015×14.00 to 210210.

\frac{7}{210}

​210

​

​7

​​  

6 Simplify.

1/30

Answer:

Sally; Sally

Step-by-step explanation:

For the free throws... let's see the stats:

Sally: 10 free throws

Jesse: 8 free throws.

Advantage?: Sally

For the three-points baskets:

Sally: 3

Jesse: 1

Advantage: Sally

Sally dominates in both categories, sorry Jesse.

Answer:

sally sally

Step-by-step explanation:

the numbers are just greater for both stats for her

Answer:

x = 52/9

Step-by-step explanation:

The exterior angle is half the difference of the intercepted arcs, so we have ...

9x -5 = (158 -64)/2

9x = 52 . . . . . . . . . . . add 5

x = 52/9 = 5 7/9

Answer:

Step-by-step explanation:

The total sold is the sum of the individual numbers sold, hence c+h.

We assume "3 times more" means "3 times as many", so the number of hotdogs sold (h) is 3 times the number of cheeseburgers sold (c), hence 3c.

  c + h = 220

  h = 3c

_____

55 cheeseburgers and 165 hotdogs were sold.

Answer:

Nickola swam for 4 hours and ran for 5.5 hours

Step-by-step explanation:

To solve this, we can use a system of equations.

First we can set up a system of equations like this

[tex]2s+15r=90.5[/tex] and

[tex]s+r=9.5[/tex]

Next we will use substitution to solve for one of the values. We can solve the second equation such that

[tex]s=9.5-r[/tex]

Now we can substitute this into the first equation for s

[tex]2(9.5-r)+15r=90.5[/tex]

Now we can solve for r

[tex]19-2r+15r=90.5[/tex]

[tex]19+13r=90.5[/tex]

[tex]13r=71.5[/tex]

[tex]r=5.5[/tex]

Now we can plug this value into the second equation to get the value for s

[tex]s+5.5=9.5[/tex]

[tex]s=4[/tex]

Now we can plug these values into the first equation to make sure we have the right values

[tex]2(4)+15(5.5)=90.5[/tex]

[tex]90.5=90.5[/tex]

Answer:

D.

Step-by-step explanation:

I also haven't learned this yet but i could tell that in the second image if A.F = 1/2AC and DE = A.F, therefore DE = 1/2AC. The problem is that i don't know if it is B or D.

Sorry .-.

Answer:

5/12 or 41.66%

Step-by-step explanation:

When throwing two six-sided dice, you have 36 possible outcomes:

{1,1} {1,2} {1,3} {1,4} {1,5} {1,6} {2,1} {2,2} {2,3} {2,4} {2,5} {2,6} {3,1} {3,2} {3,3} {3,4} {3,5} {3,6} {4,1} {4,2} {4,3} {4,4} {4,5} {4,6} {5,1} {5,2} {5,3} {5,4} {5,5} {5,6} {6,1} {6,2} {6,3} {6,4} {6,5} {6,6}

To find what is the probability the second player will win, we need to see how many of those 36 possibilities have a combined total of 8 or more (to beat the 7 of the first player):

These 15 combinations have a total of 8 or more:

{2,6} {3,5} {3,6} {4,4} {4,5} {4,6} {5,3} {5,4} {5,5} {5,6} {6,2} {6,3} {6,4} {6,5} {6,6}

So, the probability the second player gets 8 or more and wins is:

15/36 or 5/12 or 41.66%

Find the total of male students:

4 + 6 + 2 + 2 = 14 total males.

There are 2 male juniors.

The probability of a male being a junior is 2/14 = 1/7 = 0.143 = 14.3 = 14%

Find the total of male students:

4 + 6 + 2 + 2 = 14 total males.

There are 2 male juniors.

The probability of a male being a junior is 2/14 = 1/7 = 0.143 = 14.3 = 14%

Answer:

116.75 oceanic miles

Step-by-step explanation:

A graph of the data shows the distance to be between 110 and 120 miles (closer to 120). There is only one answer choice in that range.

In 2 hours, the ship travels 42.5 miles, so in 1 hour will travel 21.25 miles. Adding that distance to the distance at 4 hours gives the distance at 5 hours, ...

  95.5 +21.25 = 116.75 . . . . "oceanic" miles

_____

In order for the distance from the lighthouse to be uniformly increasing, the ship must be traveling directly away from the lighthouse. Traveling at any other angle, the distances will not fall on a straight line. (That is one reason I wanted to graph the data.)

Answer:

Associative property of multiplication

Step-by-step explanation:

To show that 9(3x) = 27(x), we need to show that 9(3x) = (9 * 3)x.

The APM does just that. By this property, in multiplication, the order of which numbers are multiplied do not matter.

So, 9(3x) = (9 * 3)x.

And by multiplication, (9 * 3)x = 27x.

So 9(3x) = 27x

Answer:

165

Step-by-step explanation:

[tex]5c^{2} -3d+15[/tex]

c = 6 and d = 10

[tex]5c^{2}[/tex] = 5 × 6² = 5 × 36 = 180

[tex]5c^{2}[/tex] - ( 3 d ) = 180 - ( 3 × 10 ) = 180 - 30 = 150

[tex]5c^{2}[/tex] -  3 d  ( + 15 ) = 150 + 15 = 165

Answer:

165

Step-by-step explanation:

Substitutet 6 for c and 10 for d in  5c^2 - 3d + 15​ .

Note that " ^ " is used here to denote exponentiation; c2 is meaningless.

Then we have 5(6)^2 - 3(10) + 15, or   180 - 30 + 15, or 165.

The ODE

[tex]M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0[/tex]

is exact if

[tex]\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}[/tex]

We have

[tex]M=-xy\sin x+2y\cos x\implies M_y=-x\sin x+2\cos x[/tex]

[tex]N=2x\cos x\implies N_x=2\cos x-2x\sin x[/tex]

so the ODE is indeed not exact.

Multiplying both sides of the ODE by [tex]\mu(x,y)=xy[/tex] gives

[tex]\mu M=-x^2y^2\sin x+2xy^2\cos x\implies(\mu M)_y=-2x^2y\sin x+4xy\cos x[/tex]

[tex]\mu N=2x^2y\cos x\implies(\mu N)_x=4xy\cos x-2x^2y\sin x[/tex]

so that [tex](\mu M)_y=(\mu N)_x[/tex], and the modified ODE is exact.

We're looking for a solution of the form

[tex]\Psi(x,y)=C[/tex]

so that by differentiation, we should have

[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]

[tex]\implies\begin{cases}\Psi_x=\mu M\\\Psi_y=\mu N\end{cases}[/tex]

Integrating both sides of the second equation with respect to [tex]y[/tex] gives

[tex]\Psi_y=2x^2y\cos x\implies\Psi=x^2y^2\cos x+f(x)[/tex]

Differentiating both sides with respect to [tex]x[/tex] gives

[tex]\Psi_x=-x^2y^2\sin x+2xy^2\cos x=2xy^2\cos x-x^2y^2\sin x+\dfrac{\mathrm df}{\mathrm dx}[/tex]

[tex]\implies\dfrac{\mathrm df}{\mathrm dx}=0\implies f(x)=c[/tex]

for some constant [tex]c[/tex].

So the general solution to this ODE is

[tex]x^2y^2\cos x+c=C[/tex]

or simply

[tex]x^2y^2\cos x=C[/tex]

We are to verify and confirm if the given differential equations are exact or not. Then solve for the exact equation.

The first differential equation says:

[tex]\mathbf{(-xy \ sin x + 2y \ cos x) dx + 2(x \ cos x) dy = 0 }[/tex]

Recall that:

A differential equation that takes the form [tex]\mathbf{M(x,y)dt + N(x, y)dy = 0 }[/tex] will be exact if and only if:

From equation (1), we can represent M and N as follows:

Thus, taking the differential of M and N, we have:

[tex]\mathbf{ \dfrac{\partial M}{\partial y }= M_y = -x sin x + 2cos x}[/tex]

[tex]\mathbf{ \dfrac{\partial N}{\partial x }= N_x = 2 cos x + 2x sin x}[/tex]

From above, it is clear that:

[tex]\mathbf{\dfrac{\partial M }{\partial y} \neq \dfrac{\partial N }{\partial x}}[/tex]

∴

We can conclude that the equation is not exact.

Now, after multiplying the given differential equation in (1) by the integrating factor μ(x, y) = xy, we have:

Representing the equation into form M and N, then:

[tex]\mathbf{M = -x^22y^2 sin x +2xy^2 cos x}[/tex]

[tex]\mathbf{N = 2x^2y cos x}[/tex]

Taking the differential, we have:

[tex]\mathbf{\dfrac{\partial M}{\partial y }= M_y = -2x^2y sin x + 4xy cos x }[/tex]

[tex]\mathbf{\dfrac{\partial N}{\partial x} =N_x= 4xycos \ x -2x^2 y sin x}[/tex]

Here;

[tex]\mathbf{\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} }[/tex]

Therefore, we can conclude that the second equation is exact.

Now, the solution of the second equation is as follows:

[tex]\int_{y } M dx + \int (not \ containing \ 'x') dy = C[/tex]

[tex]\rightarrow \int_{y } (-x^2y^2 sin(x) +2xy^2 cos (x) ) dx + \int(0)dy = C[/tex]

[tex]\rightarrow-y^2 \int x^2 sin(x) dx +2y ^2 \int x cos (x) dx = C[/tex]    ---- (3)

Taking integrations by parts:

[tex]\int u v dx = u \int v dx - \int (\dfrac{du}{dx} \int v dx) dx[/tex]

∴

[tex]\int x^2 sin (x) dx = x^2 \int sin(x) dx - \int (\dfrac{d}{dx}(x^2) \int (sin \ (x)) dx) dx[/tex]

[tex]\to x^2 (-cos (x)) \ - \int 2x (-cos \ (x)) \ dx[/tex]

[tex]\to -x^2 (cos (x)) \ + \int 2x \ cos \ (x) \ dx[/tex]   ----- replace this equation into (3)

∴

[tex]\rightarrow-y^2( -x^2 cos (x) \ + \int 2x \ cos \ (x) \ dx) +2y ^2 \int x cos (x) dx = C[/tex]

[tex]\mathbf{\rightarrow -x^2 y^2 cos (x) \ -2y ^2 \int x \ cos \ (x) \ dx +2y ^2 \int x cos (x) dx = C}[/tex]

[tex]\mathbf{x^2y^2 cos (x) = C\ \text{ where C is constant}}[/tex]

Therefore, from the explanation, we've can conclude that the first equation is not exact and the second equation is exact.

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Answer:

Eric’s share in the profit is $30,000 ⇒ answer B

Step-by-step explanation:

* We will use the ratio to solve this problem

- At first lets find the ratio between their invested

∵ Eric has invested $400,000

∵ George has invested $300,000

∵ Denzel has invested $300.000

- To find the ratio divide each number by 100,000

∴ Eric : George : Denzel = 4 : 3 : 3

- They will divide the profits among themselves in the ratio of their

  respective investments

- The total profit will divided by the total of their ratios

∵ The total of the ratios = 4 + 3 + 3 = 10

∴  Eric : George : Denzel : Sum = 4 : 3 : 3 : 10

- That means the profit will divided into 10 equal parts

- Eric will take 4 parts, George will take 3 parts and Denzel will take

 3 parts

∵ The profit = $75,000

- Divide the profit by the sum of the ratio

∴ Each part of the profit = 75,000 ÷ 10 = $7,500

- Now lets find the share of each one

∴ The share of Eric = 4 × 7,500 = $30,000

∴ The share of George = 3 × 7,500 = $22,500

∴ The share of Denzel = 3 × 7,500 = $22,500

* Eric’s share in the profit is $30,000

# If you want to check your answer add the shares of them, the answer

  will be the total profit (30,000 + 22,500 + 22,500 = $75,000), and if

  you find the ratio between their shares it will be equal the ratio

  between their investments (divide each share by 7,500 to simplify

  them the answer will be 4 : 3 : 3)

Answer:

B

Step-by-step explanation:

Answer:

3.5 seconds, B

Step-by-step explanation:

This is an upside down parabola, a function that is extremetly useful in helping us to understand position and velocity and time and how they are all related.  Her upwards velocity is 16 ft/sec and she starts from a height of 140 feet, according to the problem.  The h is the height she ends up at after a certain amount of time has gone by.  You want to know how long it will take her to hit the water.  When she hits the water, she has no more height.  Therefore, her height above the water when she hits the water is 0.  Plug in a 0 for h and factor the quadratic to get t = -2.5 seconds and t = 3.5 seconds.  The only two things in math that will never ever be negative is a distance measure and time, so we can disregard the -2.5 and go with 3.5 seconds as our answer.

Answer:

B

Step-by-step explanation:

1. C. -512√3+512i

2. B. 16(cos240°+i sin240°)

3. D. 3√2+3√6i, -3√2-3√6i

4. A. cos60°+i sin60°, cos180°+i sin180°, cos300°+i sin300°

5. D. 2√3(cos π/6+i sin π/6), 2√3(cos 7π/6+i sin 7π/6)

We will see that the equivalent expression is:

[tex]8*(cos(240\°) + i*sin(240\°))[/tex]

So the correct option is the first one.

We have the expression:

[2*(cos(240°) + i*sin(240°))]^4

Remember that Euler's formula says that:

[tex]e^{ix} = cos(x) + i*sin(x)[/tex]

Then we can rewrite our expression as:

[tex][2*(cos(240\°) + i*sin(240\°)]^4 = [2*e^{i*240\°}]^4[/tex]

Now we distribute the exponent:

[tex]2^4*e^{4*i*240\°} = 8*e^{i*960\°}[/tex]

Now, we need to find an angle equivalent to 960°.

Remember that the period of the trigonometric functions is 360°, then we can rewrite:

960° - 2*360° = 240°

This means that 960° is equivalent to 240°. Then we can write:

[tex]8*e^{i*960\°} = 8*e^{i*240\°} = 8*(cos(240\°) + i*sin(240\°))[/tex]

So the correct option is the first one.

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Expand:

[tex]n(n-1)-(n+3)(n+2)=(n^2-n)-(n^2+5n+6)=-6n-6[/tex]

Then we can write

[tex]n(n-1)-(n+3)(n+2)=6\boxed{(-n-1)}[/tex]

which means [tex]6\mid n(n-1)-(n+3)(n+2)[/tex] as required.

Answer:

Option C and D are correct.

Step-by-step explanation:

Area of rectangle = 144 cm^2

Width of rectangle = 9 cm

Length of rectangle = ?

We know,

Area of rectangle = Length * Width

144 = Length * 9

144/9 = Length

=> length = 16 cm

Option A is incorrect as 3 times width = 3* 9 = 27 but our length = 16 cm

Option B is incorrect as length = 16 cm and not 63 cm

Option C is correct as Length < 2(Width)

=> 16 < 2(9) => 16 < 18 which is true.

Option D is correct.

Perimeter = 2(Length + Width)

Perimeter = 2(16+9)

Perimeter = 50 cm

Option E is incorrect as Length ≠ Width

Answer:

C. The length is less than 2 times the width.

D. The perimeter is 50 centimeters.

Step-by-step explanation:

The area of the rectangle is given as 144 square centimeters and its width is 9 centimeters. The formula for the area of a rectangle is given as;

Area = length*width

144 = length*9

length = 144/9

length = 16 centimeters

A. The length is 3 times the width.

3 times the width; 3*9 = 27 cm which is not equal to 16. Hence this statement is false.

B.The length is 63 centimeters.

This statement is also false since the length is 16 cm

C.The length is less than 2 times the width.

Answer:

x=2

Step-by-step explanation:

The events “cheering for Team A” and “being a female” are not mutually exclusive. The probability that a randomly selected attendee is female or cheers for Team A is approximately 0.741.

(a) No, the events “cheering for Team A” and “being a female” are not mutually exclusive. This is because there are females who are cheering for Team A. Mutually exclusive events cannot happen at the same time.

(b) To find the probability that a randomly selected attendee is female or cheers for Team A, we need to add the probabilities of each event happening and subtract the probability of both events happening at the same time. We can use the formula:

P(A or B) = P(A) + P(B) - P(A and B)

In this case, P(A) is the probability of cheering for Team A, P(B) is the probability of being female, and P(A and B) is the probability of being a female who cheers for Team A.

Given the numbers provided, the probability of cheering for Team A is 2118/4997 and the probability of being female is 2568/4997. The probability of being a female who cheers for Team A is 982/4997. Plugging these values into the formula, we get:

P(Female or Team A) = P(Team A) + P(Female) - P(Female and Team A) = 2118/4997 + 2568/4997 - 982/4997 = 3704/4997 ≈ 0.741

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Answer: Third Option

[tex]f(x) +3[/tex]

Step-by-step explanation:

The function [tex]y=log_2(x)[/tex] passes through point (2, 1) because the exponential function [tex]2 ^ x = 2[/tex] when [tex]x = 1[/tex].

Then, if the transformed function passes through point (2, 4) then this means that the graph of [tex]y=log_2(x)[/tex] was moved vertically 3 units up.

The transformation that vertically displaces the graph of a function k units upwards is:

[tex]y = f (x) + k[/tex]

Where k is a positive number. In this case [tex]k = 3[/tex]

Then the transformation is:

[tex]f(x) +3[/tex]

and the transformed function is:

[tex]y = log_2 (x) +3[/tex]

Answer:

B) 34.1 cm

Step-by-step explanation:

The longer diagonal is longer than either side, but shorter than their sum. The only answer choice in the range of 24–39 cm is choice B.

_____

You are given sufficient information to use the Law of Cosines to find the diagonal length. If we call it "c", then the angle opposite that diagonal is the larger of the angles in the parallelogram: 120°. The law of cosines tells you ...

c^2 = a^2 +b^2 -2ab·cos(C)

Here, we have a=24, b=15, C=120°, so ...

c^2 = 24^2 +15^2 -2·24·15·cos(120°) = 576 +225 +360 = 1161

c = √1161 ≈ 34.073 . . . . cm

Rounded to tenths, the diagonal length is 34.1 cm.

Answer:

Step-by-step explanation:

The absolute value function prevents the expression from being a polynomial. The degree of 3 in y^3 is an odd number so that polynomial function will not be even.

Answer:

The equation [tex]y=x^3[/tex] is not an equation of a simple , even polynomial function.

Step-by-step explanation:

Even  function : A function  is even when its graph is symmetric with respect to y-axis.

Algebrically , the function f is even if and only if

f(-x)=f(x) for all x in the domain of f.

When the function does not satisfied the above condition then the function is called non even function.

f(x)[tex]\neq[/tex] f(-x)

Now , we check given function is even or not

A. y= [tex]\mid x\mid[/tex]

If x is replaced by -x

Then we get the function

f(-x)=[tex]\mid -x \mid[/tex]

f(-x)=[tex]\mid x \mid[/tex]

Hence, f(-x)=f(x)

Therefore , it is even  polynomial function.

B. [tex]y=x^2[/tex]

If x is replace by -x

Then we get

f(-x)=[tex](-x)^2[/tex]

f(-x)=[tex]x^2[/tex]

Hence, f(-x)=f(x)

Therefore, it is even polynomial function.

C. [tex]y=x^3[/tex]

If x is replace by -x

Then we get

f(-x)=[tex](-x)^3[/tex]

f(-x)=[tex]-x^3[/tex]

Hence, f(-x)[tex]\neq[/tex] f(x)

Therefore, it is not even polynomial function.

D.[tex]y= -x^2[/tex]

If x is replace by -x

Then we get

f(-x)= - [tex](-x)^2[/tex]

f(-x)=-[tex]x^2[/tex]

Hence, f(-x)=f(x)

Therefore, it is even polynomial function.

Answer: C. [tex]y=x^3[/tex] is not simple , even polynomial function.

ANSWER

The volume of the dilated cone is

[tex]40 {m}^{3}[/tex]

EXPLANATION

The volume of the given cone is

[tex]5000 {m}^{3} [/tex]

When this cone is dilated with a scale factor of 1/5, the volume of the dilated cone becomes,

[tex] ({ \frac{1}{5} })^{3} \times 5000 {m}^{3} [/tex]

We simplify to obtain:

[tex] { \frac{1}{125} }\times 5000 {m}^{3} [/tex]

This gives us:

[tex]40 {m}^{3} [/tex]

Final answer:

When a cone is scaled down by a factor of 1/5, its new volume is 40 m³, calculated by cubing the scale factor and multiplying it by the original volume.

Explanation:

When a cone is dilated by a scale factor, its volume changes according to the cube of that scale factor.

Since the original volume of the cone is 5000 m³ and the scale factor is 1/5, we use the proportionality principle which states that the volume of a shape is proportional to the cube of its linear dimensions (V ∝ L3).

Therefore, if we dilate the cone by a scale factor of 1/5, the new volume (V1) would be:

V1 = V-original × (scale factor)³
= 5000 m³ × (1/5)³
= 5000 m³ × 1/125
= 40 m³

This calculation shows that, as a result of applying the scale factor, the volume of the cone has been reduced significantly.

The answer is:

The number of teeth of Gear 2 is 90 teeth.

[tex]N_{2}=90teeth[/tex]

To calculate the number of teeth for the Gear 2, we need to use the following formula that establishes a relation between the number of RPM and the number of teeth of two or more gears.

[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]

Where,

N, are the rpm of the gears

Z, are the teeth of the gears.

We are given the following information:

[tex]Z_{1}=30teeth\\N_{1}=150RPM\\N_{2}=50RPM[/tex]

Then, substituting and calculating we have:

[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]

[tex]150RPM*30teeth=N_{2}50RPM[/tex]

[tex]N_{2}=\frac{150RPM*30teeth}{50RPM}=90teeth[/tex]

[tex]N_{2}=90teeth[/tex]

Hence, we have that the number of teeth of Gear 2 is 90 teeth.

Have a nice day!

Answer:

P(cream and sugar) = 0.4

Step-by-step explanation:

* Lets study the meaning of or , and on probability

- The use of the word or means that you are calculating the probability

 that either event A or event B happened

-  Both events do not have to happen

- The use the word and, means that both event A and B have to happen

* The addition rules are:

# P(A or B) = P(A) + P(B) ⇒ mutually exclusive (events cannot happen

 at the same time)

# P(A or B) = P(A) + P(B) - P(A and B) ⇒ non-mutually exclusive (if they

 have at least one outcome in common)

- The union is written as A∪B or “A or B”.

- The intersection is written as A∩B or “A and B”

* Lets solve the question

∵ P(cream) = 0.5

∵ P(sugar) = 0.6

∵ P(cream or sugar) = 0.7

- To find P(cream and sugar) lets use the rule of non-mutually exclusive

∵ P(A or B) = P(A) + P(B) - P(A and B)

∴ P(cream or sugar) = P(cream) + P(sugar) - P(cream and sugar)

- Lets substitute the values of P(cream) , P(sugar) , P(cream or sugar)

 in the rule

∵ 0.7 = 0.5 + 0.6 - P(cream and sugar) ⇒ add the like terms

∴ 0.7 = 1.1 - P(cream and sugar) ⇒ subtract 1.1 from both sides

∴ 0.7 - 1.1 = - P(cream and sugar)

∴ - 0.4 = - P(cream and sugar) ⇒ multiply both sides by -1

∴ 0.4 = P(cream and sugar)

* P(cream and sugar) = 0.4

Answer:

0.4

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

The volume of a sphere is given by

[tex]V=\frac{4}{3}\pi r^3[/tex]

where r is the radius

For $2.75, we can get 1 large  OR  2 small scoops.

Giant scoop has diameter 6, so radius is half of that, which is 3, hence the volume is:

[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (3)^3\\V=113.1[/tex]

Regular scoop's diameter is 4, hence radius is 2. So volume of 1 regular scoop is:

[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (2)^3\\V=33.51[/tex]

We can get 2 of those, so total volume is 33.51 + 33.51 = 67.02

Hence, the max volume for $2.75 is around 113, answer choice B.

ANSWER

[tex]y = 2x -4[/tex]

EXPLANATION

Part a)

Eliminating the parameter:

The parametric equation is

[tex]x = 5 + ln(t) [/tex]

[tex]y = {t}^{2} + 5[/tex]

From the first equation we make t the subject to get;

[tex]x - 5 = ln(t) [/tex]

[tex]t = {e}^{x - 5} [/tex]

We put it into the second equation.

[tex]y = { ({e}^{x - 5}) }^{2} + 5[/tex]

[tex]y = { ({e}^{2(x - 5)}) } + 5[/tex]

We differentiate to get;

[tex] \frac{dy}{dx} = 2 {e}^{2(x - 5)} [/tex]

At x=5,

[tex] \frac{dy}{dx} = 2 {e}^{2(5 - 5)} [/tex]

[tex]\frac{dy}{dx} = 2 {e}^{0} = 2[/tex]

The slope of the tangent is 2.

The equation of the tangent through

(5,6) is given by

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y - 6 = 2(x - 5)[/tex]

[tex]y = 2x - 10 + 6[/tex]

[tex]y = 2x -4[/tex]

Without eliminating the parameter,

[tex] \frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } [/tex]

[tex]\frac{dy}{dx} = \frac{ 2t}{ \frac{1}{t} } [/tex]

[tex]\frac{dy}{dx} = 2 {t}^{2} [/tex]

At x=5,

[tex]5 = 5 + ln(t) [/tex]

[tex] ln(t) = 0[/tex]

[tex]t = {e}^{0} = 1[/tex]

This implies that,

[tex]\frac{dy}{dx} = 2 {(1)}^{2} = 2[/tex]

The slope of the tangent is 2.

The equation of the tangent through

(5,6) is given by

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y - 6 = 2(x - 5) =[/tex]

[tex]y = 2x -4[/tex]

The equation of the tangent to the curve at the point (5,6) is [tex]\(y = 2x - 4\)[/tex].

To find the equation of the tangent to the curve given by the parametric equations [tex]\(x = 5 + \ln(t)\)[/tex] and [tex]\(y = t^2 + 5\)[/tex] at the point (5,6), we can approach this problem in two ways: by eliminating the parameter \(t\) and without eliminating the parameter.

Method 1: Eliminating the Parameter

Step 1: Express (t) in terms of (x)

[tex]\[ x = 5 + \ln(t) \implies \ln(t) = x - 5 \implies t = e^{x-5} \][/tex]

Step 2: Substitute (t) into (y)

[tex]\[ y = t^2 + 5 \implies y = (e^{x-5})^2 + 5 \implies y = e^{2(x-5)} + 5 \][/tex]

Step 3: Find [tex]\(\frac{dy}{dx}\)[/tex]

[tex]\[ y = e^{2(x-5)} + 5 \][/tex]

[tex]\[ \frac{dy}{dx} = 2e^{2(x-5)} \][/tex]

Step 4: Evaluate [tex]\(\frac{dy}{dx}\)[/tex] at (x = 5)

[tex]\[ \frac{dy}{dx}\bigg|_{x=5} = 2e^{2(5-5)} = 2e^0 = 2 \][/tex]

Step 5: Equation of the tangent line

The slope (m = 2). The tangent line at (5,6) is:

[tex]\[ y - 6 = 2(x - 5) \][/tex]

[tex]\[ y = 2x - 10 + 6 \][/tex]

[tex]\[ y = 2x - 4 \][/tex]

Method 2: Without Eliminating the Parameter

Step 1: Find [tex]\(\frac{dx}{dt}\)[/tex] and [tex]\(\frac{dy}{dt}\)[/tex]

[tex]\[ x = 5 + \ln(t) \implies \frac{dx}{dt} = \frac{1}{t} \][/tex]

[tex]\[ y = t^2 + 5 \implies \frac{dy}{dt} = 2t \][/tex]

Step 2: Find [tex]\(\frac{dy}{dx}\)[/tex]

[tex]\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2t}{\frac{1}{t}} = 2t^2 \][/tex]

Step 3: Find (t) at the point (5,6)

From [tex]\(x = 5 + \ln(t)\)[/tex]:

[tex]\[ 5 = 5 + \ln(t) \implies \ln(t) = 0 \implies t = e^0 = 1 \][/tex]

Step 4: Evaluate [tex]\(\frac{dy}{dx}\)[/tex] at (t = 1)

[tex]\[ \frac{dy}{dx}\bigg|_{t=1} = 2(1)^2 = 2 \][/tex]

Step 5: Equation of the tangent line

The slope (m = 2). The tangent line at (5,6) is:

[tex]\[ y - 6 = 2(x - 5) \][/tex]

[tex]\[ y = 2x - 10 + 6 \][/tex]

[tex]\[ y = 2x - 4 \][/tex]

Thus, using both methods, the equation of the tangent to the curve at the point (5,6) is [tex]\(y = 2x - 4\)[/tex].