The amount of money that is left in a medical savings account is expressed by the equation y = negative 24 x + 379, where x represents the number of weeks and y represents the amount of money, in dollars, that is left in the account. After how many weeks will the account have $67 left in it? 10 weeks 13 weeks 15 weeks 21 weeks

Answer:

The inverse of the function is [tex]f^{-1}(x)=\frac{x-5}{3}[/tex].

Step-by-step explanation:

The function provided is:

[tex]f (x)=3x+5[/tex]

Let [tex]f(x)=y[/tex].

Then the value of x is:

[tex]y=3x+5\\\\3x=y-5\\\\x=\frac{y-5}{3}[/tex]

For the inverse of the function, [tex]x\rightarrow y[/tex].

⇒ [tex]f^{-1}(x)=\frac{x-5}{3}[/tex]

Compute the value of [tex]f[f^{-1}(x)][/tex] as follows:

[tex]f[f^{-1}(x)]=f[\frac{x-5}{3}][/tex]

               [tex]=3[\frac{x-5}{3}]+5\\\\=x-5+5\\\\=x[/tex]

Hence proved that [tex]f[f^{-1}(x)]=x[/tex].

Compute the value of [tex]f^{-1}[f(x)][/tex] as follows:

[tex]f^{-1}[f(x)]=f^{-1}[3x+5][/tex]

               [tex]=\frac{(3x+5)-5}{3}\\\\=\frac{3x+5-5}{3}\\\\=x[/tex]

Hence proved that [tex]f^{-1}[f(x)]=x[/tex].

Answer:

six x squared plus seven

Step-by-step explanation:

um..

Answer:

The way to write 6x² + 7 in words is six x squared plus seven

Step-by-step explanation:

I hope your happy with your answer

Answer:

a = 30

b = 6/7

Step-by-step explanation:

The number of yeast cells after t hours is modeled by the following equation:

[tex]f(t) = a(1 + be^{-0.7t})[/tex]

In which a is the initial number of cells.

At time t = 0 the population is 30 cells

This means that [tex]a = 30[/tex]

So

[tex]f(t) = 30(1 + be^{-0.7t})[/tex]

And increasing at a rate of 18 cells/hour.

This means that f'(0) = 18.

We use this to find b.

[tex]f(t) = 30(1 + be^{-0.7t})[/tex]

So

[tex]f(t) = 30 + 30be^{-0.7t}[/tex]

Then, it's derivative is:

[tex]f'(t) = -30*0.7be^{-0.7t}[/tex]

We have that:

f'(0) = 18

So

[tex]f'(0) = -30*0.7be^{-0.7*0} = -21b[/tex]

Then

[tex]-21b = 18[/tex]

[tex]21b = -18[/tex]

[tex]b = -\frac{18}{21}[/tex]

[tex]b = \frac{6}{7}[/tex]

Answer:

(fog)(x) means that we have the function f(x) evaluated in the function g(x), or f(g(x)).

So, if f(x) = x^3 + 1 and g(x) = x - 2.

we have:

a) (fog)(0) = f(g(0)) = (0 - 2)^3 + 1 = -8 + 1 = -7

b) (gof)(0) = g(f(0)) = (0^3 + 1) - 2 = -1

c)  (fof)(1) = f(f(1)) = (1^3 + 1)^3 + 1 = 2^3 + 1 = 8 + 1 = 9

d) (gog)(1) = g(g(1)) = (1 - 2) - 2 = -1 -2 = -3

Answer:

9 bags

Step-by-step explanation:

130, 134, 136, 145, 145, 147, 147, 151, 154

9 bags had at least 130 peanuts.

Answer:

The message is S  E  C R  E  T     A G E N T       M A N

Step-by-step explanation:

This is a very easy exercise, follow the steps taken below:

Step 1: Find [tex]A^{-1}[/tex]

[tex]A = \left[\begin{array}{ccc}1&2\\1&3\end{array}\right][/tex]

Note that If [tex]M = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex]

then [tex]M^{-1} = \left[\begin{array}{ccc}a&-b\\-c&a\end{array}\right][/tex]

Therefore, [tex]A^{-1} = \left[\begin{array}{ccc}3&-2\\-1&1\end{array}\right][/tex]

Step 2: Find A⁻¹B ( i.e product of A and B)

[tex]B = \left[\begin{array}{cccccccc}29&39&45&2&17&54&26&29\\34&57&65&3&22&74&39&43\end{array}\right][/tex]

[tex]A^{-1} B = \left[\begin{array}{ccc}3&-2\\-1&1\end{array}\right] \left[\begin{array}{cccccccc}29&39&45&2&17&54&26&29\\34&57&65&3&22&74&39&43\end{array}\right][/tex]

[tex]A^{-1}B = \left[\begin{array}{cccccccc}19&3&5&0&7&14&0&1\\5&18&20&1&5&20&13&14\end{array}\right][/tex]

Step 3: Decode the message in the matrix above

The message will be decoded in this order:

Note that 0 stands for  empty space

19 5 3 18 5 20 0 1 7 5 14 20 0 13 1 14

S  E  C R  E  T     A G E N T       M A N

The message encoded is:  S  E  C R  E  T     A G E N T       M A N

Answer:

55.82 cm

Step-by-step explanation:

d1= 8.72 cm

a= 15.6 cm

A rhombus= 1/2*d1*d2 = A square

A square= 15.6²= 243.36 cm²

d2= 2A/d1= 2*243.36/8.72 ≈55.82 cm

Answer:

"a quantity x over two plus 4 times the quantity x." would be your answer.

Answer:

D = 300 m

Step-by-step explanation:

Given :

Speed = 5 m/s

Time = 1 min = 60 secs

Required:

Distance = ?

Formula :

Speed = Distance/Time

Solution:

Distance = Speed × Time

D = 5 × 60

D = 300 m

Total distance walked = 5 x 60 (there are 60 seconds in a minute)

                                     = 300 metres

Answer:

The discriminant is zero which means there is 1 repeated real solution

Step-by-step explanation:

9x^2 -12x +4 = 0

This is in the form

ax^2 +bx +c

a = 9 b=-12 c=4

The discriminant is

b^2 -4ac

(-12)^2 - 4(9)(4)

144 -144

0

The discriminant is zero which means there is 1 repeated real solution

Answer:

Area = 108 cm^2

Perimeter = 44 cm

Step-by-step explanation:

Area, -->

24 + 30 + 24 + 30 -->

24(2) + 30(2)

48 + 60 = 108 cm^2

108 = area

10 + 12 + 10 + 12, -->

10(2) + 12(2) = 44 cm

44 = perim.

Hope this helps!

Answer:

Step-by-step explanation:

Draw the diagram.

This time put in the only one line for the height. That is only 1 height is 8 cm. That's it.

The base is 6 +  6 = 12 cm.

The slanted line is 10 cm

That's all your diagram should show. It is much clearer without all the clutter.

Now you are ready to do the calculations.

Area

The Area = the base * height.

base = 12

height = 8

Area = 12 * 8 = 96

Perimeter.

In a parallelagram the opposite sides are equal to one another.

One set of sides = 10 + 10 = 20

The other set = 12 + 12 = 24

Both sets = 20 + 24

Both sets = 44

Answer

Area = 96

Perimeter = 44

Answer:

66.7%

Step-by-step explanation:

The factors of 18 are as listed: 1, 2, 3, 6, 9, 18

Since 1, 2, 3, and 6 are in the die, that makes 4 out of the 6 numbers.

4/6 can be simplified to .667 or 66.7%

Answer:

66.7%

Step-by-step explanation:

Factors of 18: 1, 2, 3, 6, 9, 18

There are 4 factors of 18 on the die so there is a 4/6 chance or 2/3 chance rolling a factor of 18. As a percent, 66.7%

Answer:

See below for explanation

Step-by-step explanation:

A related question can be found at chegg. Find attached the diagram.

The triangle is a right angled triangle. The 3 sides are named opposite, adjacent and hypotenuse.

The opposite is the vertical side facing angle theta (θ)

The adjacent is horizontal side

The angle between opposite and adjacent = 90°

The hypotenuse is the longest side

The relationship between the 3 sides:

Hypotenuse² = opposite ² + adjacent²

The relationship between theta (θ)

and the lengths of the sides of the triangle:

Applying SOHCAHTOA in trigonometry

Sine ratio:

Sinθ = opposite/hypotenuse

Cosine ratio:

Cosθ = adjacent/hypotenuse

Tangent ratio:

Tanθ = opposite/adjacent

Applying the Pythagorean theorem to find the third side of the triangle

Hypotenuse² = opposite ² + adjacent²

Hypotenuse = 7, opposite = 4

adjacent² = Hypotenuse² - opposite ²

adjacent² = 7²-4² = 49-16

adjacent² = 33

Adjacent = √33

Write out the six trigonometric functions in exact form related to theta.

cosθ = adjacent/hypotenuse = (√33)/7

sinθ = opposite/hypotenuse = 4/7

tanθ = sinθ/cosθ = opposite/adjacent

sinθ/cosθ = 4/7 ÷ (√33)/7

= 4/7 × 7/(√33) = 4/(√33)

opposite/adjacent = 4/(√33)

tanθ = 4/(√33)

secθ = 1/cosθ = 1/(√33)/7

= 7/√33

cosecθ = 1/sinθ = 1/(4/7)

= 1×7/4 = 7/4

cotθ = 1/tanθ = 1/[4/(√33)]

cotθ =(√33)/4

Answer:

(x-7)^2 + (y-2) ^2 = 130

Step-by-step explanation:

We need to find the length of the radius

the length of the radius is found by using the distance formula

r = sqrt(  ( -2-7)^2+( -5 -2)^2)

    sqrt(  ( -9)^2+( -7)^2)

    sqrt( 81+49)

    sqrt(130)

The formula for a circle is

(x-h)^2 + (y-h) ^2 = r^2

where (h,k) is the center and r is the radius

(x-7)^2 + (y-2) ^2 = 130

The Laplace transform of the function [tex]\frac{1}{2} (\frac{1}{s} - \frac{s}{s^2 + 4w^2} )[/tex] .

The Laplace transform exist when s > 0 .

Here, the given function is f(t) = sin²(wt) .

The Laplace transform of the the function f(t),

F(s) = f(t) = { [tex]{\frac{1}{2} \times 2sin^2(wt) }[/tex] }

F(s) = { [tex]\frac{1}{2} \times (1- cos2wt)[/tex] }

F(s) = { [tex]\frac{1}{2} - \frac{1}{2} \times cos(2wt)\\[/tex] }

F(s) = [tex]\frac{1}{2} (\frac{1}{s} - \frac{s}{s^2 + 4w^2} )[/tex]

Next,

The above Laplace transform exist if s > 0 .

Know more about Laplace transform,

https://brainly.com/question/31481915

#SPJ4

Answer:

(a)0.34

(b)0.5

(c)0.185

(d)0.544

(e)0.655

(f)Not disjoint events

(g)Independent events

Step-by-step explanation:

The summary of education levels among party lines is given in the table below

[tex]\left|\begin{array}{c|ccc}&$Democrat &$Republican &$Total\\--&--&--&--\\$College Degree &37 &31 &68\\$No College Degree& 63 &69 &132\\--&--&--&--\\$Total &100 &100& 200\end{array}\right|[/tex]

(a)Probability a randomly selected participant has a college degree.

The probability a randomly selected participant has a college degree is:

[tex]=\dfrac{68}{200} =0.34[/tex]

(b)The probability that a randomly selected participant is a democrat.

[tex]=\dfrac{100}{200} =0.5[/tex]

(c)The probability that a randomly selected participant is a democrat and has a college degree.

[tex]=\dfrac{37}{200} =0.185[/tex]

(d)Probability of being a democrat given that the participant has a college degree.

[tex]=\dfrac{37}{68} \approx 0.544[/tex]

(e)The probability of begin a democrat or having a college degree

Number of Democrats =100

Number of Those with College degrees = 68

Number of democrats with College degrees =37

Therefore:

Probability of begin a democrat or having a college degree

[tex]=\dfrac{100+68-37}{200} \\=\dfrac{131}{200}\\\\=0.655[/tex]

(f)The event of having a college degree and being a democrat are not disjoint events as there are some democrats who have a college degree.

(g)The event of having a college degree and being a democrat are independent events as the outcome of one does not affect the other.

Answer:

Step-by-step explanation:

This problem could keep you going for quite a while. My suggestion is that you go get a cup of coffee and sip it slowly as you read this.

Equation One

Sqrt(x - 1)^3 = 8

(x - 1)^(3/2) = 8

Square both sides to get rid of the 2.

(x - 1)^3 = 8^2  

(x - 1)^3 = 64

Now take the cube root of both sides to get rid of the 3 on the left

x - 1 = cuberoot(64)

x - 1 = 4                     Add 1 to both sides

x - 1+1 = 4 + 1

x  =  5

==============================

Second Equation

4th root (x - 3)^5 = 32

Take the 5th root of both sides.

4th root(x - 3) = 2

This can be written as (x - 3)^(1/4) = 2

Now take the 4th power of both sides.

(x - 3) = 2^4

x - 3 =  16

add 3 to both sides.

x = 16 + 3

x = 19

============================

Equation 3

(x - 4)^(3/2) = 125

Take the cube root of both sides

(x - 4)^(1/2) = 125^(1/3)     1/3 is the cube root of something

(x - 4)^(1/2) = 5    

square both sides to get rid of the 2    

(x - 4) = 5^2

x - 4 = 25

Add 4 to both sides.

x = 25 + 4

x = 29

============================

Fourth Equation

(x + 2)^(4/3) = 16        

take the 4th root of both sides

(x + 2) ^(1/3) = 16^(1/4)

(x + 2)^(1/3) = 2

Cube both sides

(x + 2) = 2^3

x + 2 = 8  

Subtract 2 from both sides

x + 2 - 2 = 8-2

x = 6

##########################

The first step is the most critical. You must look at what you are going to take the root of. When you do, for this question, it must come out even.

Answer:

[tex]y = - [ \frac{1}{t - \frac{28}{9} } + t ][/tex]

Step-by-step explanation:

Solution:-

- A change of variable is a technique employed in solving many differential equations that are of the form: y ' = f ( t , y ).

- Considering a differential equation of the form y' = f ( αt + βy + γ ), where α, β, and γ are constants. A substitution of an arbitrary variable z = αt + βy + γ is made and the given differential equation is converted into a form: z ' = g ( z ).

- This substitution basically allow us to solve in-separable differential equations by converting them into a form that can be separated, followed by the set procedure.

- We are to solve the initial value problem for the following differential equation:

                              [tex]y' = ( t + y ) ^2 - 1 , y ( 3 ) = 6[/tex]

First Step: Make the appropriate substitution

- We will use a arbitrary variable ( z ) and define the our substitution by finding a multi-variable function f ( t , y ) that is a part of the given ODE.

- We see that the term ( t + y ) is a multi-variable function and also the culprit that doesn't allow us to separate our variables.

- Usually, the change of variable substitution is made for such " culprits ".

- So our substitution would be:

                               [tex]z = t + y[/tex]

Second Step: Implicit differential of the substitution variable ( z ) with respect to the independent variable

- In the given ODE we see that the variable ( t ) is our independent variable. So we will derivate the supposed substitution as follows:

                              [tex]\frac{dz}{dt} = 1 + \frac{dy}{dt} \\\\\frac{dy}{dt} = -1 + \frac{dz}{dt}[/tex]

Remember: z is a multivariable function of "t" and "y". So we perform implicit differential for the variable " z ".

Third Step: Plug in the differential form in step 2 and change of variable substitution of ( z ) in the given ODE.

- The given ODE can be expressed as:

                            [tex]\frac{dy}{dt} = ( t + y ) ^2 - 1\\\\\frac{dz}{dt} - 1 = ( z ) ^2 - 1\\\\\frac{dz}{dt} = z ^2 \\[/tex] ... Separable ODE

Fourth Step: Separate the variables and solve the ODE.

- We see that the substitution left us with a simple separable ODE.

Note: If we do not arrive at a separable ODE, then we must go back and re-choose our change of variable substitution for ( z ).

- We will progress by solving our ODE:

                            [tex]\frac{dz}{z^2} = dt\\\\\int {\frac{1}{z^2} } \, dz = \int {1} \, dt\\\\-\frac{1}{z} = t + c\\\\\frac{1}{z} = - (t + c )\\\\z = -\frac{1}{t + c}[/tex]

Where,

                     c: The constant of integration

Fifth Step: Back-substitution of variable ( z )

- We will now back-substitute the substitution made in the first step and arrive back at our original variables ( y and t ) as follows:

                           [tex]t + y = - \frac{1}{t + c} \\\\y = - [ \frac{1}{t + c} + t ][/tex]  

Sixth Step: Apply the initial value problem and solve for the constant of integration ( c )

- We will use the given initial value statement i.e y ( 3 ) = 6 and evaluate the constant of integration ( c ) as follows:

                           [tex]y ( 3 ) = - [ \frac{1}{3 + c} + 3 ] = 6 \\\\\frac{1}{3 + c} = -9\\\\3 + c = -\frac{1}{9} \\\\c = - \frac{28}{9}[/tex]

Seventh Step: Express the solution of the ODE in an explicit form ( if possible ):

                          [tex]y = - [ \frac{1}{t - \frac{28}{9} } + t ][/tex]

Answer:

[tex]x\le \:9[/tex]

Step-by-step explanation:

[tex]5x\le 45[/tex]

[tex]\frac{5x}{5}\le \frac{45}{5}[/tex]

[tex]x\le \:9[/tex]

Answer:

B

Step-by-step explanation:

We divide the entire inequality by 5 to get rid of the coefficient of x. The ≤ stays the same so we get x ≤ 9.

Answer:

r^-8s^-5 = 1/r^8 * s^5 = s^5/r^8

s^5/r^8 = r^-8s^5

Owen forgot put s^5 in the denominator to make its exponent negative.

Answer: D

Step-by-step explanation:

Sometimes, it is hard to see which function fits g(x). We are given the point (4,1). Luckily, the point lies on g(x). We can plug in (4,1) into the g(x) equations and see which best fits.

A. Incorrect

[tex]1=4(4)^2[/tex]

[tex]1=4(16)[/tex]

[tex]1\neq 64[/tex]

B. Incorrect

[tex]1=\frac{1}{4} (4)^2[/tex]

[tex]1=\frac{1}{4} (16)[/tex]

[tex]1\neq 4[/tex]

C. Incorrect

[tex]1=(\frac{1}{2}(4))^2[/tex]

[tex]1=(2)^2[/tex]

[tex]1\neq 4[/tex]

D. Correct

[tex]1=(\frac{1}{4}(4))^2[/tex]

[tex]1=(1)^2[/tex]

[tex]1=1[/tex]

Answer:

[tex]P(t) = P(0)e^{0.0693t}[/tex]

Step-by-step explanation:

The population of a community is known to increase at a rate proportional to the number of people present at time t.

This means that the population growth is modeled by the following differential equation:

[tex]\frac{dP(t)}{dt} = rP(t)[/tex]

Which has the following solution:

[tex]P(t) = P(0)e^{rt}[/tex]

In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.

The population has doubled in 10 years

This means that [tex]P(10) = 2P(0)[/tex]. We use this to find r.

[tex]P(t) = P(0)e^{rt}[/tex]

[tex]2P(0) = P(0)e^{10r}[/tex]

[tex]e^{10r} = 2[/tex]

[tex]\ln{e^{10r}} = \ln{2}[/tex]

[tex]10r = \ln{2}[/tex]

[tex]r = \frac{\ln{2}}{10}[/tex]

[tex]r = 0.0693[/tex]

So the equation that will estimate the population of the community in t years is:

[tex]P(t) = P(0)e^{0.0693t}[/tex]

Answer:

41 word/min

Step-by-step explanation:

Before noon Ali works:

She types:

After lunch she works:

She types:

Total Ali works= 4+4= 8 hours= 480 min

Total Ali types= 11520+8160= 19680 words

Average typing rate= 19680 words/480 min= 41 word/min

Answer:

x<16

Step-by-step explanation:

number n

eight less than four times a number ... 4 x n - 8

is less than 56 ... < 56

4 x n - 8 < 56

4 x n < 56 + 8

4 x n < 64/4

n < 64 / 4

n < 16

Answer:

Step-by-step explanation:

Let the number be x

Four times the number :  4x

Eight less than four times a number: 4x - 8

4x - 8 < 56

Now add 8 to both sides,

4x < 56+8

4x < 64

Divide both sides by 4,

x < 64/4

x < 16

Possible values of number = Value less than 16

Answer:

[tex]220-2.01\frac{12}{\sqrt{50}}=216.589[/tex]    

[tex]220+2.01\frac{12}{\sqrt{50}}=223.411[/tex]    

The confidence interval for this case would be (216.589, 223.411)

Step-by-step explanation:

Information given

[tex]\bar X=220[/tex] represent the sample mean

[tex]\mu[/tex] population mean

s=12 represent the sample standard deviation

n=50 represent the sample size  

Confidence interval

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

The degrees of freedom are given by:

[tex]df=n-1=50-1=49[/tex]

The Confidence level is 0.95 or 95%, the significance is [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], the critical value for this case is [tex]t_{\alpha/2}=2.01[/tex]

And replacing we got:

[tex]220-2.01\frac{12}{\sqrt{50}}=216.589[/tex]    

[tex]220+2.01\frac{12}{\sqrt{50}}=223.411[/tex]    

The confidence interval for this case would be (216.589, 223.411)

Answer:

The probability of finding five​ left-handers in the sample is P(x=5)=0.094.

Step-by-step explanation:

We can model this variable with a binomial distribution, with parameters n=46 (sample size) and p=0.170 (the probability that an individual is​ left-handed).

The probability that k individuals are left-handed in the sample can be calculated as:

[tex]P(x=k) = \dbinom{n}{k} p^{k}(1-p)^{n-k}\\\\\\P(x=k) = \dbinom{46}{k} 0.17^{k} 0.83^{46-k}\\\\\\[/tex]

Then, the probability of finding five​ left-handers in the sample is:

[tex]P(x=5) = \dbinom{46}{5} 0.17^{5}\cdot 0.83^{41}=1370754*0.000142*0.000481=0.094\\\\\\[/tex]

Answer:

[tex]a. \quad 8\left(x+a\right)\\\\b. \quad 8\left(2x+h\right)[/tex]

Step-by-step explanation:

Best Regards!

Answer:  (a) 8(x  + a)   -->    8x + 8a

               (b) 8(2x + h)  -->   16x + 8h  

Step-by-step explanation:

f(x) = 8x²

f(a) = 8a²

[tex]\dfrac{f(x)-f(a)}{x-a}\quad = \quad \dfrac{8x^2-8a^2}{x-a}\quad = \quad \dfrac{8(x-a)(x+a)}{x-a}=\large\boxed{8(x+a)}[/tex]

f(x + h) = 8(x + h)²

           = 8(x² + 2xh + h²)

           = 8x² + 16xh + 8h²

f(x) = 8x²

[tex]\dfrac{f(x+h)-f(x)}{h} = \dfrac{(8x^2+16xh+8h^2)-8x^2}{h}\\\\\\.\qquad \qquad \qquad \quad =\dfrac{16xh + 8h^2}{h}\\\\\\.\qquad \qquad \qquad \quad =\dfrac{8h(2x + h)}{h}\\\\\\.\qquad \qquad \qquad \quad =\large\boxed{8(2x+h)}[/tex]

Answer: Store B

Step-by-step explanation:

180 / 3 = 60. 180 - 60= $120. Store A cost is $120.

110 * 0.9 = $99. Store B's cost is $99.

Answer:

Store B

Step-by-step explanation:

Store A the price would be about $120.60

Store B price would be about $99

To find store a price, you first find the discount, so

0.33 x 180 = 59.40

Then subtract this from the original price to know the total after the discount

180-59.40=120.60

Do the same thing with the other Store

110 x 0.10 = 11

110-11=99

Answer:

1.2kg

Step-by-step explanation:

6 identical toys weigh 1.8kg.

1 toy would weigh:

1.8/6 = 0.3

0.3 kg.

Multiply 0.3 with 4 to find how much 4 identical toys would weigh.

0.3 × 4 = 1.2

4 identical toys would weigh 1.2kg

Answer:

[tex]1.2kg[/tex]

Step-by-step explanation:

6 identical toys weigh = 1.8kg

Let's find the weight of 1 toy ,

[tex]1.8 \div 6 = 0.3[/tex]

Now, lets find the weigh of 6 toys,

[tex]0.3 \times 4 = 1.2kg[/tex]