The water park has trolley cars that visitors use to ride throughout the park. The total number of people that all the cars can hold is greater than 480 and less than 500 people. (a) what is a possible number of trolley cars in the park and a possible number of people in each car if all the cars are being used? Each car has the same number of people. Both the number of trolley cars and the number of people in each car must have two digits and be less than 30. (b) explain how you found your answer. (c) what is another solution for the problem in part (a)?

One marker would cost approx. $0.03

Answer:

[tex]\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}[/tex]

Step-by-step explanation:

We are given a function:

[tex]\displaystyle \large{f(x) = x^2 \sin (x)}[/tex]

Notice that there are two functions multiplying each other. Recall the product rules.

[tex]\displaystyle \large{y=h(x)g(x) \to y\prime = h\prime (x) g(x) + h(x)g\prime (x)}[/tex]

You can let x^2 = h(x), sin(x) = g(x) or sin(x) = h(x), x^2 = g(x) as your desire but I’ll let h(x) = x^2 and g(x) = sin(x).

Therefore, from a function:

[tex]\displaystyle \large{f(x) = x^2 \sin (x) \to f\prime (x) = (x^2)\prime \sin(x) + x^2 (\sin (x))\prime}[/tex]

Recall the power rules and differentiation of sine.

[tex]\displaystyle \large{y = ax^n \to y\prime = nax^{n-1} \ \ \ \tt{for \ \ polynomial \ \ function}}\\ \displaystyle \large{y = \sin (x) \to y\prime = \cos (x) }[/tex]

Therefore, from differentiating function,

[tex]\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}[/tex]

And we are done!

Find where the equation is undefined ( when the denominator is equal to 0.

Since they say x = 5, replace x in the equation see which ones equal o:

5-5 = 0

So we know the denominator has to be (x-5), this now narrows it down to the first two answers.

To find the horizontal asymptote, we need to look at an equation for a rational function: R(x) = ax^n / bx^m, where n is the degree of the numerator and m is the degree of the denominator.

In the equations given neither the numerator or denominators have an exponent ( neither are raised to a power)

so the degrees would be equal.

Since they are equal the horizontal asymptote is the y-intercept, which is given as -2.

This makes the first choice the correct answer.

Hi, here is the solution.

The doubling each dimensions increases the volume by 8. That is (2 * 2* 2)

There 1000 toothpicks in a regular box.

The jumbo box hold  8 *1000 = 8000 toothpicks.

Thank you.

Answer:

2x-3y=20

12x+8y=-192

Solution

In the determinant of the matrix to find the variable "x" [Ax], we replace the coefficients of "x" (2 and 12) by the independent terms (20 and -192 respectively).

In the determinant of the matrix to find the variable "y" [Ay], we replace the coefficients of "y" (-3 and 8) by the independent terms (20 and -192 respectively).

Marshall can use x = 8 and y = 0 to show that 8² = 64 using the polynomial identity (x−y)^2=x^2−2xy+y^2.

Marshall can use any values for x and y to show that (x−y)^2=x^2−2xy+y^2. However, he specifically wants to show that 8² = 64. In this case, Marshall can choose x = 8 and y = 0.

Using the polynomial identity (x−y)^2=x^2−2xy+y^2, we substitute x = 8 and y = 0 to get (8 − 0)^2 = 8^2 − 2(8)(0) + 0^2. Simplifying, we have 8^2 = 64 − 0 + 0, which simplifies further to 8^2 = 64.

So, Marshall can use the values x = 8 and y = 0 to show that 8² = 64 using the given polynomial identity.

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Answer:

This equation is a line

Step-by-step explanation:

We can tell it's a line because both the x and y values are raised to the first power. In order for it to be any of the others, one or both of the variables would have to have a non-one exponent.

Answer:

It is a line.

Step-by-step explanation:

Since, the general equation of,

A line is ax + by = c

A circle is [tex](x-h)^2+(y-k)^2=r^2[/tex]

An ellipse is [tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]

A parabola is [tex]y^2=4ax[/tex] or [tex]x^2=4ay[/tex]

A hyperbola is [tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]

Where, a, b and c are constants.

Here, the given equation is,

x + y = 5

Which is the type of ax + by = c

Hence, x + y = 5 is a line.

y = | 3x - 7 |

x-intercept is when y = 0

0 = | 3x - 7 |

0 = 3x - 7

+7       +7

7 = 3x

[tex]\frac{7}{3} = \frac{3x}{3}[/tex]

 [tex]\frac{7}{3} = x[/tex]

[tex](\frac{7}{3}, 0 )[/tex]

y-intercept is when x = 0

y = | 3(0) - 7 |

y = | 0 - 7 |

y = | -7 |

y = 7

(0, 7)

Answer: [tex](\frac{7}{3}, 0 )[/tex] and (0, 7)

It seems that you are trying to find the length of hair of Florencia on Tuesday.

Given that length of the hair of Florencia on Monday = h centimeters

On Tuesday, she got hair cut.

So new length of the hair of Florencia on Tuesday = 75% of Monday value

= 75% of (h)

= 0.75* (h)

= 0.75h

Hence final answer is the length of the hair of Florencia on Tuesday is ( 0.75h ) centimeters.

Answer:

3/4h and 0.75h

Given equation is [tex]x^3-5x^2+2x+6=0[/tex]

Usually we solve this type of problem by factoring but this equation is not factorable. If we solve by grouping then also we are not getting any common factor. So factoring method is not going to help. I this type of situation we may use graphing calculator. From graph we just need to pick those points where graph crosses the x-axis. Those values will be the final answer.

Using graphing calculator we see that there are three solutions at x=-0.856, x=1.678, x=4.177 which are approx x=-0.8, x=1.7 and x=4.2

Hence final answers are x=-0.8, x=1.7 and x=4.2.

Answer:

-0.9, 1.7, and 4.2

Step-by-step explanation:

7×2=14e

7×1=7

14e-7-11=6+6e

   -6e             -6e

8e-7-11=6

    +7      +7

8e-11=13

     +11    +11

8e=24

8e÷8=e

24÷8=3

e=3

To solve the equation 7(2e-1)-11=6+6e for e, distribute, combine like terms, subtract, add, and divide until you have isolated e. The solution is e=3.

To solve for e in the equation 7(2e-1)-11=6+6e, first distribute the 7 on the left side of the equation:

14e - 7 - 11 = 6 + 6e

Now, combine like terms on both sides:

14e - 18 = 6 + 6e

Next, subtract 6e from both sides:

8e - 18 = 6

Add 18 to both sides:

8e = 24

Finally, divide by 8:

 e = 3.

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To solve an absolute value equation of the form |X| = k, where X is an expression with a variable and k is a non-negative number or an expression, you must solve the compound equation X = k or X = -k.

The first step is to divide both sides by 9 to get the absolute value of an expression equal to a non-negative number.

9|9 – 8x| = 2x + 3

Divide both sides by 9.

[tex] |9 - 8x| = \dfrac{2x + 3}{9} [/tex]

Now we eliminate the absolute value by rewriting this as a compound equation.

[tex] 9 - 8x = \dfrac{2x + 3}{9} ~~~or~~~ 9 - 8x = -\dfrac{2x + 3}{9} [/tex]

We must solve both equations.

Multiply both sides by 9.

[tex] 9(9 - 8x) = 2x + 3 ~~~or~~~ 9(9 - 8x) = -(2x + 3) [/tex]

[tex] 81 - 72x = 2x + 3 ~~~or~~~ 81 - 72x = -2x - 3 [/tex]

[tex] -74x = -78 ~~~or~~~ -70x = -84 [/tex]

[tex] x = \dfrac{-78}{-74} ~~~or~~~ x = \dfrac{-84}{-70} [/tex]

[tex] x = \dfrac{39}{37} ~~~or~~~ x = \dfrac{6}{5} [/tex]

Answer: x = 39/37 or x = 6/5

To check for extraneous solutions, we plug in each solution into the original equation and check if it works.

Check 39/37:

9|9 – 8(39/37)| = 2(39/37) + 3

|81 - 72(39/37)| = 78/37 + 3

|2997/37 - 2808/37| = 78/37 + 111/37

|189/37| = 189/37

189/37 = 189/37

The solution 39/37 is valid.

Check x = 6/5.

9|9 – 8x| = 2x + 3

9|9 – 8(6/5)| = 2(6/5) + 3

|81 - 72(6/5)| = 12/5 + 15/5

|405/5 - 432/5| = 27/5

|-27/5| = 27/5

27/5 = 27/5

The solution x = 6/5 is also valid.

There are no extraneous solutions.

You can subtract that height of volcano which lies below sea level from the total height to obtain height of volcano above sea level.

Thus, height of specified volcano is 4214 meters.

The height of volcano above sea level

You can subtract that height of volcano which lies below sea level from the total height to obtain height of volcano above sea level.

Thus:

Height of volcano above sea level = total height of volcano - depth of volcano below sea level

[tex]\text{Height of volcano above sea level} = 10207 - 5993 = 4,214 \: \rm meters[/tex]

Thus, height of specified volcano is 4214 meters.

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d = r * t

                               time                rate                       distance

Downstream:           2                   8 + c                     d = 2(8 + c)

Upstream:                3                   8 - c                      d = 3(8 - c)

Since the distance is the same, we can set the distance equations equal to each other and solve for c.

2(8 + c) = 3(8 - c)

16 + 2c = 24 - 3c

16 + 5c = 24

      5c = 8

        c = [tex]\frac{8}{5}[/tex]

        c = 1.6

Answer: 1.6 mph

8,5 and 6,1

The slope is 2

To factor a number out of an expression means to divide each term of the expression by that number.

So, if you want to factor [tex] a [/tex] from the expression [tex] x+y+z [/tex], you'll write it as

[tex] a\left(\dfrac{x}{a}+\dfrac{y}{a}+\dfrac{z}{a}\right) [/tex]

So, in your case, when you divide both terms by -5 you get

[tex] -10a \div (-5) = 2a,\quad -25\div (-5) = 5 [/tex]

So, the expression becomes

[tex] -10a-25 = -5(2a+5) [/tex]

Answer:

Option B 0.11

Step-by-step explanation:

Given that  the population proportion of 0.58, a simulation of 200 trials is run, each with a sample size of 50 and a point estimate of 0.42. The minimum sample proportion from the simulation is 0.31, and the maximum sample proportion from the simulation is 0.53.

This gives an idea that confidence interval is

(0.31, 0.53) with mean being in the middle of this interval

Margin of error = 1/2 (length of this interval)

Length of confidence interval =[tex]0.53-0.31 =0.22[/tex]

Margin of error =± 1/2(0.22) = ±0.11

Hence option B is correct

In this mathematical context, 'f' represents a vehicle's fuel efficiency in miles per gallon, and the given inequality f > 40 mpg indicates that the vehicle's fuel efficiency is higher than 40 miles per gallon.

The problem refers to a situation where a vehicle achieves fuel efficiency of more than 40 miles per gallon. This efficiency is quantified using the variable 'f'.

Considering f as the vehicle's fuel efficiency in miles per gallon, you are expressing a relationship where f > 40 mpg.

Using an example: if we consider a vehicle that achieves 43 miles per gallon, this can be represented as: f = 43 mpg. This efficiency is indeed more than 40 miles per gallon, thus confirming that f > 40 mpg in this case. It's important to understand that in this context, the fuel efficiency varies from one vehicle to another. For instance, vehicles become more fuel efficient over the years due to advancements in technology and heightened fuel economy standards.

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4, 12, 16, 48, 52, 156, 160, ______, _______, _______

x3  +4  x3  +4  x3    +4

That pattern is: multiply by 3, add 4, multiply by 3, add 4, ...

160 x 3 = 480

480 + 4 = 484

484 x 3 = 1452

Answer: 480, 484, 1452

The next 3 numbers are 480, 484, 1452

The given sequence is:

4,12,16,48,52,156,160,_,_,_

The rule of the sequence is:

The last three terms are the 7th, 8th, and 9th terms

The 7th term is:

[tex]T_7=3T_6\\\\T_7=3(160)\\\\T_7=480[/tex]

The 8th term is:

[tex]T_8=T_7+4\\\\T_8=480+4\\\\T_8=484[/tex]

The 9th term is:

[tex]T_9=3T_8\\\\T_9=3(484)\\\\T_9=1452[/tex]

Therefore, the next 3 numbers are 480, 484, 1452

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Step 1

Find the measure of angle x

we know that

If ray NP bisects <MNQ

then

m<MNQ=m<PNM+m<PNQ ------> equation A

and

m<PNM=m<PNQ -------> equation B

we have that

m<MNQ=(8x+12)°

m<PNQ=78°

so

substitute in equation A

(8x+12)=78+78-------> 8x+12=156------> 8x=156-12

8x=144------> x=18°

Step 2

Find the measure of angle y

we have

m<PNM=(3y-9)°

m<PNM=78°

so

3y-9=78------> 3y=87------> y=29°

therefore

the answer is

the measure of x is 18° and the measure of y is 29°

Answer: If NP bisects MNQ, MNQ=8x+12,PNQ=78, and RNM=3y-9, find the values of x and y

x= 18 y=11

Step-by-step explanation:

We know that MNQ= 8x+12

PNQ=78 and MNP is equal to PNQ

Therefore, PNQ=MNP

So since these two angles make up MNQ,

Add 78+78 which is 156

So now we need to find x

The equation to find x is 8x+12=156

8x+12=156

8x=144

x=18

Now that we know x it is time to find y.

So we know that RNM=3y-9

And we know that MNQ is equal to 156

RNM an MNQ form a straight line which means that it is equal to 180 degrees

So the equation to find y is 3y-9+156=180

3y-9+156=180

3y+147=180

3y=33

y=11

So in conclusion, x=18 and y=11

Hope this helped! :3

4x³ - 20x² + 24x = 0

4x(x² - 5x + 6) = 0

4x(x - 2)(x - 3) = 0

4x = 0        x - 2 = 0          x - 3 = 0

 x = 0            x = 2              x = 3

Answer: 0, 2, 3  

Factoring and solving a quadratic equation, it is found that the roots of the equation are:

[tex]x = 0[/tex]

[tex]x = 2[/tex]

[tex]x = 3[/tex]

The equation given is:

[tex]4x^3 - 20x + 24x = 0[/tex]

Factoring the common term:

[tex]4x(x^2 - 5x + 6) = 0[/tex]

For the roots:

[tex]4x = 0 \rightarrow x = 0[/tex]

Or

[tex]x^2 - 5x + 6 = 0[/tex]

Which is a quadratic equation with coefficients [tex]a = 1, b = -5, c = 6[/tex], and then:

[tex]\Delta = (-5)^{2} - 4(1)(6) = 1[/tex]

[tex]x_{1} = \frac{-(-5) + \sqrt{1}}{2} = 3[/tex]

[tex]x_{2} = \frac{-(-5) - \sqrt{1}}{2} = 2[/tex]  

Thus, the other two roots are [tex]x = 2[/tex] and [tex]x = 3[/tex].

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