A bus company has contracted with a local high school to carry 450 students on a field trip. The company has 18 large buses which can carry up to 30 students and 19 small buses which can carry up to 15 students. There are only 20 drivers available on the day of the field trip.
The total cost of operating one large bus is $225 a day, and the total cost of operating one small bus is $100 per day.
​

(1) I assume "log" on its own refers to the base-10 logarithm.

[tex]\left(e^{-2x}\log(\ln x)^3\right)'=\left(e^{-2x}\right)'\log(\ln x)^3+e^{-2x}\left(\log(\ln x)^3\right)'[/tex]

[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{e^{-2x}}{\ln10(\ln x)^3}\left((\ln x)^3\right)'[/tex]

[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}(\ln x)^2}{\ln10(\ln x)^3}\left(\ln x\right)'[/tex]

[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}(\ln x)^2}{\ln10\,x(\ln x)^3}[/tex]

[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}}{\ln10\,x\ln x}[/tex]

Note that writing [tex]\log(\ln x)^3=3\log(\ln x)[/tex] is one way to avoid using the power rule.

(2)

[tex]\left(e^{-2x}(\log(\ln x))^3\right)'=(e^{-2x})'(\log(\ln x))^3+e^{-2x}\left(\log(\ln x))^3\right)'[/tex]

[tex]=-2e^{-2x}(\log(\ln x))^3+3e^{-2x}(\log(\ln x))^2(\log(\ln x))'[/tex]

[tex]=-2e^{-2x}(\log(\ln x))^3+3e^{-2x}(\log(\ln x))^2\dfrac{(\ln x)'}{\ln10\,\ln x}[/tex]

[tex]=-2e^{-2x}(\log(\ln x))^3+\dfrac{3e^{-2x}(\log(\ln x))^2}{\ln10\,x\ln x}[/tex]

(3)

[tex]\left(\sin(xe^x)^3\right)'=\left(\sin(x^3e^{3x})\right)'=\cos(x^3e^{3x}(x^3e^{3x})'[/tex]

[tex]=\cos(x^3e^{3x})((x^3)'e^{3x}+x^3(e^{3x})')[/tex]

[tex]=\cos(x^3e^{3x})(3x^2e^{3x}+3x^3e^{3x})[/tex]

[tex]=3x^2e^{3x}(1+x)\cos(x^3e^{3x})[/tex]

(4)

[tex]\left(\sin^3(xe^x)\right)'=3\sin^2(xe^x)\left(\sin(xe^x)\right)'[/tex]

[tex]=3\sin^2(xe^x)\cos(xe^x)(xe^x)'[/tex]

[tex]=3\sin^2(xe^x)\cos(xe^x)(x'e^x+x(e^x)')[/tex]

[tex]=3\sin^2(xe^x)\cos(xe^x)(e^x+xe^x)[/tex]

[tex]=3e^x(1+x)\sin^2(xe^x)\cos(xe^x)[/tex]

(5) Use implicit differentiation here.

[tex](\ln(xy))'=(e^{2y})'[/tex]

[tex]\dfrac{(xy)'}{xy}=2e^{2y}y'[/tex]

[tex]\dfrac{x'y+xy'}{xy}=2e^{2y}y'[/tex]

[tex]y+xy'=2xye^{2y}y'[/tex]

[tex]y=(2xye^{2y}-x)y'[/tex]

[tex]y'=\dfrac y{2xye^{2y}-x}[/tex]

Answer:

2.8 cm

Step-by-step explanation:

The map scale is 1 cm : 5 km. That means that 1 cm is equal to 5 km.

To find the map distance (in cm), we have to set up a ratio.

[tex]\frac{1 cm}{5 km} = \frac{x}{14 km}[/tex]

X (the map distance in cm) is over the actual distance of 14 km.

Now cross multiply and divide.

[tex]5x = 14[/tex]

[tex]\frac{5x}{5} = \frac{14}{5}[/tex]

[tex]x = 2.8 cm[/tex]

If the actual distance to be represented is 14 km, the map distance (in cm) will be 2.8 cm.

Hope that helps.

Answer:

A =625 ft^2

Step-by-step explanation:

The perimeter of a square is

P = 4s where s is the side length

100 =4s

Divide each side by 4

100/4 = 4s/4

25 = s

A = s^2 for a square

A = 25^2

A =625

Answer:

x + 2y = 6

2y = -x + 6

y = -1/2x + 3

So, you will have a downward sloping, less steep line with an intercept at (0, 3).

You can use the Math is Fun Function Grapher and Calculator to graph the line.

Hope this helps!

Answer:

The p-value is 0.809.

Step-by-step explanation:

In this case we need to perform a significance test for the standard deviation.

The hypothesis is defined as follows:

H₀: σ₀ = 4 vs. Hₐ: σ₀ ≤ 4

The information provided is:

n = 9

s = 3

Compute the Chi-square test statistic as follows:

[tex]\chi^{2}=\frac{(n-1)s^{2}}{\sigma_{0}^{2}}[/tex]

     [tex]=\frac{(9-1)\cdot (3)^{2}}{(4)^{2}}\\\\=\frac{8\times 9}{16}\\\\=4.5[/tex]

The test statistic value is 4.5.

The degrees of freedom is:

df = n - 1

   = 9 - 1

   = 8

Compute the p-value as follows:

[tex]p-value=P(\chi^{2}_{9}>4.5)=0.809[/tex]

*Use a Chi-square table.

Thus, the p-value is 0.809.

Answer:

(3) x ≥ -3

(4) 2.5 gallons

(4) -12x + 36

Step-by-step explanation:

Hey there!

1)

Well its a solid dot meaning it will be equal to.

So we can cross out 1 and 2.

And it's going to the right meaning x is greater than or equal to -3.

(3) x ≥ -3

2)

Well if each milk container has 1 quart then there is 10 quarts.

And there is 4 quarts in a gallon, meaning there is 2.5 gallons of milk.

(4) 2.5 gallons

4)

16 - 4(3x - 5)

16 - 12x + 20

-12x + 36

(4) -12x + 36

Hope this helps :)

Answer:

Option D.

Step-by-step explanation:

The vertex form of an absolute function is

[tex]y=a|x-h|+k[/tex]

where, a is a constant, (h,k) is vertex.

It is given that, vertex of an absolute function is (2,3). So, h=2 and k=3.

[tex]y=a|x-2|+3[/tex]   ...(1)

It crosses the x-axis at (5,0). So put x=5 and y=0 to find the value of a.

[tex]0=a|5-2|+3[/tex]

[tex]-3=3a[/tex]

[tex]-1=a[/tex]

Put a=-1 in (1).

[tex]y=(-1)|x-2|+3[/tex]

[tex]y=-|x-2|+3[/tex]

Now, put y=0, to find the equation for this absolute value function when y=0.

[tex]0=-|x-2|+3[/tex]

Therefore, the correct option is D.

Answer:

I got this question on my test and I answered D cause if you look up the graph it matches the question

Step-by-step explanation:

D 0=−|x−2|+3

Answer:

7.81

Step-by-step explanation:

its a triangular shape

let x = 4 + 2 = 6

let y = 5

length between two points = h

h² = x² + y²

h² = 6² + 5²

h = sqrt of 61

h = 7.81

The probability that the shirt that is chosen at random from the drawer is not a white shirt, can be found to be  47 %

The probability that the shirt picked is not a white shirt can be found by first finding the number of shirts that are not white shorts in the drawer. This number is :

= Number of black shirts + Gray shirts

= 3 + 4

= 7 shirts

Then, find the total number of shirts in the drawer, including the white shirts :
=  Number of black shirts + Gray shirts  + White shirts

= 3 + 8 + 4

= 15 shirts

The probability that when a shirt is chosen at random, that it is not a white shirt is :

= Number of shirts that are not white / Total number of shirts

= 7 / 15

= 47 %

Find out more on probability at https://brainly.com/question/27831410

#SPJ1

Answer:

[tex]\boxed{m = -2}[/tex]

Step-by-step explanation:

[tex]0 = 4+4(m+1)[/tex]

Resolving Parenthesis

[tex]0 = 4+4m + 4[/tex]

[tex]0 = 4m+8[/tex]

Subtracting 8 to both sides

[tex]-8 = 4m[/tex]

[tex]4m = -8[/tex]

Dividing both sides by 4

m = -8/4

m = -2

Step-by-step explanation:

4+4m+4= 0

4m+8=0

4m=-8

m= -8/4=-2

Answer:

31 m

Step-by-step explanation:

v=l*w*h since it is a cube then all sides (a) are equal:

v=(a*a*a)=a^3

v1+v2=1331 for the first two boxes(

a³+a³=∛1331

l=w=h=11*2=22

v=729 ( for the second cube)

a=∛729=9

9+22=31 m

Answer:

[tex] y'' =12x^2 -72=0[/tex]

And solving we got:

[tex] x=\pm \sqrt{\frac{72}{12}} =\pm \sqrt{6}[/tex]

We can find the sings of the second derivate on the following intervals:

[tex] (-\infty<x< -\sqrt{6}) , y'' = +[/tex] Concave up

[tex]x=-\sqrt{6}, y =-180[/tex] inflection point

[tex] (-\sqrt{6} <x< \sqrt{6}), y'' =-[/tex] Concave down

[tex]x=\sqrt{6}, y=-180[/tex] inflection point

[tex] (\sqrt{6}<x< \infty) , y'' = +[/tex] Concave up

Step-by-step explanation:

For this case we have the following function:

[tex] y= x^4 -36x^2[/tex]

We can find the first derivate and we got:

[tex] y' = 4x^3 -72x[/tex]

In order to find the concavity we can find the second derivate and we got:

[tex] y'' = 12x^2 -72[/tex]

We can set up this derivate equal to 0 and we got:

[tex] y'' =12x^2 -72=0[/tex]

And solving we got:

[tex] x=\pm \sqrt{\frac{72}{12}} =\pm \sqrt{6}[/tex]

We can find the sings of the second derivate on the following intervals:

[tex] (-\infty<x< -\sqrt{6}) , y'' = +[/tex] Concave up

[tex]x=-\sqrt{6}, y =-180[/tex] inflection point

[tex] (-\sqrt{6} <x< \sqrt{6}), y'' =-[/tex] Concave down

[tex]x=\sqrt{6}, y=-180[/tex] inflection point

[tex] (\sqrt{6}<x< \infty) , y'' = +[/tex] Concave up

Ans   k = 4

Step-by-step explanation:

Here g(x) =[tex]\frac{-1}{3}x + 1[/tex] and

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

Now,  g(x) = f(x) + k

    or,      [tex]\frac{-1}{3}x + 1[/tex]  =  [tex]\frac{-1}{3} x -3 + k[/tex]

    or,      1 + 3 = k

    So,  k = 4   Answer.

Answer: 9/7 or -5/2

Step-by-step explanation:

We can only factorise quadratics if they're in the format ax^2 + bx + c

Re-arranging the equation gives 14x^2 + 17x - 45 = 0

Factorising this quadratic gives:

(7x - 9)(2x + 5) = 0

There are numerous ways to factorise quadratics, using a calculator or via alternate methods you may have learnt in class. (E.g. 2 numbers multiply to make (14 * -45) and add up to make (17).

This gives us our solutions.

x = 9/7 or x = -5/2

Answer:

See below.

Step-by-step explanation:

First, move all the terms to one side so we have only a zero on the right:

[tex]6x^2-17x+13=20x^2-32\\-14x^2-17x+45=0\\14x^2+17x-45=0[/tex]

(I divided everything by negative 1 in the third step. This is optional, but I like having the first term positive.)

Now, we just need to factor it. To factor, what you want to do is find two numbers a and b such that:

When a and b is multiplied together, they equal the first coefficient and constant multiplied together.

And when a and b is added together, they equal the second term.

In other words, we want to find two numbers that when multiplied equals 14(-45)=-630 and when added equals 17. Then, we can substitute this into the 17. You do this by guessing and checking. It's useful to have a calculator.

After a bit, you can find that 35 and -18 works. Thus:

[tex]14x^2+17x-45=0\\14x^2+35x-18x-45=0\\7x(2x+5)-9(2x+5)=0\\(7x-9)(2x+5)=0[/tex]

Now, the finish the problem, we just need to use the Zero Product Property and solve for x:

[tex]2x+5=0\\x=-5/2\\\\7x-9=0\\x=9/7[/tex]

Note: This only works for quadratics.

Answer:

Step-by-step explanation:

y = -5x² - 4x + 1 is a quadratic and thus is defined on the domain "all real numbers."  Because of the negative sign in front of the x^2 term, we know that this parabolic curve opens downward.  The x-coordinate of the vertex is x = -b/[2a], which in this case is x = 4/[2*-5], or -4/10, or -2/5.  Using synthetic division to determine the y-coordinate of the vertex, we get vertex (-2/5, 9/5).  9/5 is the maximum y value.  The range is (-infinity, 9/5].

Answer:

[tex]Circumference = 21.99 \ m[/tex]

Step-by-step explanation:

Circumference = [tex]\pi d[/tex]

Given that d = 7 m

[tex]Circumference = (3.14)(7)\\[/tex]

[tex]Circumference = 21.99 \ m[/tex]

Answer:

[tex]\boxed{21.98 \: \mathrm{meters}}[/tex]

Step-by-step explanation:

Apply formula for circumference of a circle.

[tex]C=\pi d[/tex]

[tex]d:diameter[/tex]

Take [tex]\pi =3.14[/tex]

Plug [tex]d=7[/tex]

[tex]C=3.14 \times 7[/tex]

[tex]C= 21.98[/tex]

Answer: c(x) = $50*x + $24

Step-by-step explanation:

First, this situation can be modeled with a linear equation like:

c(x) = s*x + b

where c is the cost, s is the slope, x is the number of cubic yards of mulch bought, and b is the y-intercept ( a constant that no depends on the number x)

Then we know that:

The company charges $50 per cubic yard, so the slope is $50

A delivery charge of $24, this delivery charge does not depend on x, so this is the y-intercept.

Then our equation is:

c(x) = $50*x + $24

This is:

"The cost of buying x cubic yards of mulch"

Answer:

7/24

Step-by-step explanation:

7/8×3/8= 21/72

divide using 3

= 7/24

Answer:

11 + Mai's Score = 59

Step-by-step explanation:

You need to add 11 and Mai's score together to get 59, so with the values given we can make the equation 11 + Mai's Score = 59.

*depending on the question, Mai's score may need to be said as a letter variable, so:

If m = mai's score,

11 + m = 59

I hope this helped! :)

Answer:

1. Sometimes

2. Sometimes

3. Always

4. Sometimes

Step-by-step explanation:

1. Quadratic function : in which maximum power of [tex]x[/tex] is two.

The roots of quadratic function can be either equal or different.

For example:

So, sometimes is the correct answer.

2. Cubic function has atleast 1 real root.

Cubic function has maximum power of [tex]x[/tex] as 3.

If the coefficients are real numbers then atleast 1 real root.

If the coefficients are imaginary in nature, then this is not true.

For example:

Cubic equation [tex]x^3 +i = 0[/tex] does not have any real root.

Cubic equation [tex]x^3 +1 = 0[/tex] has a real root x = -1.

So, it is sometimes true.

3. A function with degree 5 i.e. maximum power of [tex]x[/tex] as 5 will have 5 roots.

It is always true that a function will have number of roots equal to its degree.

4. Quadratic function can have only 1 complex solution.

Two complex solutions are also possible for a quadratic function.

For example:

[tex]x^{2} +1=0[/tex] will have two imaginary roots: [tex]x=i, -i[/tex]

It is also possible to have 1 complex solution,

For example:

[tex](x-1)(x-i) = 0[/tex] will have one complex root and one real root.

So, the statement is sometimes true.

Answer:

MY ANSWER IS CORRECT IN PLATO!!!

1. Sometimes

2. Always

3. Always

4. Never

Step-by-step explanation:

1. A quadratic function has 2 distinct roots SOMETIMES

2. A cubic Function has at least 1 root ALWAYS

3. A function with a degree of 5 has 5 roots ALWAYS

4. A quadratic function can have only 1 complex solution NEVER

I JUST GOT 100% on the quiz in PLATO

Answer:

It does not have an answer as 3x != 3x + 13 or not equalivalent

Step-by-step explanation:

Answer:

no solution

Step-by-step explanation:

3x+8=9+3x-14

Combine like terms

3x+8 = 3x -5

Subtract 3x from each side

8 = -5

This is never true so there is no solution

Answer:

The size square removed from each corner = 32.15 cm²

Step-by-step explanation:

The volume of the box = Length * Breadth * Height

Let r be the size  removed from each corner

Note that at maximum volume, [tex]\frac{dV}{dr} = 0[/tex]

The original length of the cardboard is 119 cm, if you remove a  size of r (This typically will be the height of the box)  from the corner, since there are two corners corresponding to the length of the box, the length of the box will be:

Length, L = 119 - 2r

Similarly for the breadth, B = 24 - 2r

And the height as stated earlier, H = r

Volume, V = L*B*H

V = (119-2r)(24-2r)r

V = r(2856 - 238r - 48r + 4r²)

V = 4r³ - 286r² + 2856r

At maximum volume dV/dr = 0

dV/dr = 12r² - 572r + 2856

12r² - 572r + 2856 = 0

By solving the quadratic equation above for the value of r:

r = 5.67 or 42

r cannot be 42 because the  size removed from the corner of the cardboard cannot be more than the width of the cardboard.

Note that the area of a square is r²

Therefore, the size square removed from each corner = 5.67² = 32.15 cm²

Answer:

[tex]\frac{1}{(\sqrt[3]{x} )^5}[/tex]

Step-by-step explanation:

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

Answer:

EB = 9

Step-by-step explanation:

CD = AB

The line with the value of five that also forms a right angle with EB is a perpendicular bisector to AB.

So the value of EB is half of AB (AB is equal to CD).

18/2 = 9

Answer: A, (0, -3)

Step-by-step explanation:

Inequalities, once graphed, take the form of the image you attached:

Linear inequalities are straight lines, sometimes dotted and sometimes solid, with shading on one side of the line.

Any point in the shading is a correct solution to the inequality.

In this case, the line is solid, so any point on the line is a solution to the inequality.

Looking at answer choice A: (0, -3), it lies on the line as the y-intercept.

The correct choice is A.

Answer: 430

Step-by-step explanation:

An average of 5 scores can be found via: (the sum of the scores)*5.  Thus, simply multiply 86*5 to get that the sum of his scores is 430

Hope it helps <3

Answer:

[tex]\large \boxed{\sf \ \ v_0=32 \ \ }[/tex]

Step-by-step explanation:

Hello,

The equation is

[tex]y=f(x)=-16x^2+v_0 \cdot x[/tex]

The ball hits the ground 2 seconds after the player kicks it, it means that f(2)=0.

We need to find [tex]v_0[/tex]  such that f(2)=0.

[tex]f(2)=-16\cdot 2^2+v_0 \cdot 2=-64+2v_0=0\\\\\text{*** add 64 to both sides ***}\\\\2v_0=64\\\\\text{*** divide by 2 both sides ***} \\\\v_0=\dfrac{64}{2}=32[/tex]

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

Answer:

v0 = 32 ft/s

Step-by-step explanation:

Answer:

A)Option D

B)P(X = 15) = 0.1325

Step-by-step explanation:

A) From the question, the information given follows binomial distribution because there are two mutually exclusive outcomes for each​ trial, there is a fixed number of trials. The outcome of one trial does not affect the outcome of ​another, and the probability of success is the same for each trial.

So option D is correct.

B) From the question, we are told that the poll reported that 66 percent of adults were satisfied with the job. Thus, probability is; p = 0.66

Let X be the number of adults satisfied with the job. Since 25 are selected,

Thus;

P(X = 15) = C(25, 15) * (0.66)^(15) * (1 - 0.66)^(25 - 15)

P(X = 15) = 3268760 × 0.00196407937 × 0.00002064378

P(X = 15) = 0.1325

Answer:

Step-by-step explanation:

Given f(x, y, z) = x sin(yz), the formula for calculating the gradient of the function is expressed as ∇f(x, y, z) = fx(x, y, z)i+ fy(x, y, z)j+fz(x, y, z)k where;

fx, fy and fz are the differential of the functions with respect to x, y and z respectively.

a) ∇f(x, y, z) = sin(yz)i + xzcos(yz)j + xycos(yz)k

The gradient of f =  sin(yz)i + xzcos(yz)j + xycos(yz)k

b) Directional derivative of f at (1,2,0) in the direction of v = i + 4j − k is expressed as ∇f(1, 2, 0)*v

∇f(1, 2, 0) = sin(2(0))i +1*0cos(2*0)j + 1*2cos(2*0)k

∇f(1, 2, 0) = sin0i +0cos(0)j + 2cos(0)k

∇f(1, 2, 0) = 0i +0j + 2k

Given v = i + 4j − k

∇f(1, 2, 0)*v (note that this is the dot product of the two vectors)

∇f(1, 2, 0)*v =  (0i +0j + 2k)*(i + 4j − k )

Given i.i = j.j = k.k =1 and i.j=j.i=j.k=k.j=i.k = 0

∇f(1, 2, 0)*v = 0(i.i)+4*0(j.j)+2(-1)k.k

∇f(1, 2, 0)*v = 0(1)+0(1)-2(1)

∇f(1, 2, 0)*v =0+0-2

∇f(1, 2, 0)*v= -2

Hence, the directional derivative of f at (1, 2, 0) in the direction of v = i + 4j − k is -2

Answer:  D.) The second difference is constant.

Step-by-step explanation:

The rate of change of a quadratic function is a linear function. The rate of change of that is constant, so second differences of a quadratic number pattern are constant.

Answer:

D.

Step-by-step explanation: