Your company is introducing a fruit drink packaged in an aluminum box with a square
base. Find the surface area of this box as a function of its dimension of its base, S, given
that volume of the box is 36 in. Graph this function and determine the dimensions that
produce a minimum surface area for this aluminum box.​

Answer:

3/10

Step-by-step explanation:

P(red) = red marbles / total marbles

We have 30 red marbles

total marbles = (20 white+ 50 blue+ 30 red) = 100 marbles

P(red) = 30/100 = 3/10

Answer:

0.30 or 30%.

Step-by-step explanation:

This = number of re marbles / total number of marbles

= 30 / (30+50+20)

= 30 / 100

= 0.30.

Answer:

Step-by-step explanation:

We know:

1. Diagonals of a rhombus are perpendicular.

2. Diagonals divide the rhombus on four congruent right triangles.

3. The sum of measures of acute angles in a right triangle is equal 90°.

Therefore we have the equation:

(2x + 3) + (3x + 2) = 90           combine like terms

(2x + 3x) + (3 + 2) = 90

5x + 5 = 90               subtract 5 from both sides

5x = 85             divide both sides by 5

x = 17

The value of x is 17. This means that the angles formed by the diagonals of the rhombus are 37 degrees (2 * 17 + 3) and 53 degrees (3 * 17 + 2), and they indeed form congruent right triangles as required in the properties of a rhombus.

To find the value of x in the given equation, we start by understanding the properties of a rhombus and how its diagonals divide it into four congruent right triangles.

Given information:

Diagonals of a rhombus are perpendicular.

Diagonals divide the rhombus into four congruent right triangles.

Let's proceed with the steps to solve for x:

Step 1: Recognize that the angles formed by the diagonals are 2x + 3 and 3x + 2. Since these angles are congruent right angles, their sum is equal to 90 degrees.

Step 2: Set up the equation:

(2x + 3) + (3x + 2) = 90

Step 3: Combine like terms:

5x + 5 = 90

Step 4: Isolate x by subtracting 5 from both sides of the equation:

5x = 85

Step 5: Solve for x by dividing both sides by 5:

x = 85 / 5

x = 17

Hence, the value of x is 17. This means that the angles formed by the diagonals of the rhombus are 37 degrees (2 * 17 + 3) and 53 degrees (3 * 17 + 2), and they indeed form congruent right triangles as required in the properties of a rhombus.

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Answer:

1.6 < x < 9.6.

Step-by-step explanation:

Now x cannot be less than  5.6 - 4.0 = 1.6 as a triangle  could not be formed if it were.

In fact x must be greater than 1.6.

Also x must be less than the sum of the other 2 sides.

So x  must be less than 4.0 + 5.6 = 9.6.

Answer:

1.6 < x < 9.6

Step-by-step explanation:

The range of possible sizes for side x is 1.6 < x < 9.6.

Answer:

(-6,-3)

Step-by-step explanation:

The explanation is in the picture. I really feel like the picture does a better job of explaining it then I could with words.

Answer:

C (-6,-3)

Step-by-step explanation:

A(2,5),B(2,-3), and D(-6,5)

AB  has a length of  8  since the x coordinate is the same , we only worry about the y

5--3 = 5+3 = 8

AD has a length of 8 since the y coordinate is the same, we only worry about the x

2 --6 = 2+6 =8

Notice a pattern.

A and B have the same X

A and D have the same Y

B and C will need the same Y for it to be a square

C and D will need the same x

B has a y coordinate of -3  

D has an x coordinate of -6

C will have coordinates of (-6,-3)

If we look at the attached graph of the 3 points, we see that to make the square, we need to add a point at (-6,-3)

Answer:

B,D, and E

Step-by-step explanation:

They all label the vertices in the correct order and do not label sides when naming it.

Answer:

B. [tex]\Delta TAX[/tex],

D. [tex]\Delta AXT[/tex]

E. [tex]\Delta XTA[/tex].

Step-by-step explanation:

We have been given a triangle. We are asked to choose the valid names for our triangle from the given choices.

We know that labels of the vertices of the triangle are used to name a triangle. In naming triangle, we can start from any vertex and we should keep the letters in order as we go around the triangle.

We can see that vertices of our given triangle are labeled as A, T and X, therefore, we can get three names for our triangle as:

[tex]\Delta TAX[/tex], [tex]\Delta AXT[/tex] and [tex]\Delta XTA[/tex].

Therefore, options B, D and C are correct choices.

Answers:

1. 147+13=160

2. -55+(-31)= -55-31=-86

positive number and negative number= -negative number

3. 18+71=89

4. -14+21=7

Positive 7 because 21 is greater than -14

5. 12+(-56)= 12-56=-44

6. -4+18=14

7. -31+(-17)=-31-17=-48

8. 72+(-22)=72-22=50

9. 47+23=70

Answer:

60x^3

Step-by-step explanation:

Given:

4/15x^3 and 5/12x^2

finding LCM of denominators (15x^3,12x^2)

for variable x, as x^3 is the heighest power so we'll keep x^3

now for the numeric part

15= 3*5

12=3*4

common factor is 3, so

3*4*5 =60

hence LCD= 60x^3!

Answer:

The graph in the attached figure

Step-by-step explanation:

we have

[tex]y< x^{2} -3[/tex]

The solution of the inequality is the shaded area below the dotted line of the quadratic equation [tex]y=x^{2} -3[/tex]

using a graphing tool

see the attached figure

The graph of the given function y < x² - 3 is attached below.

we have inequality that is a mathematical statement that compares two values or expressions using a relational operator, such as less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), or not equal to (≠).

Since we are given the inequality as;

y < x² - 3

We can write as;

y = x² - 3

These are used to describe relationships between quantities or to express constraints or conditions.

Therefore, the solution to the inequality is the shaded area below the dotted line of the quadratic equation.

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Answer:

32,292,000

Step-by-step explanation:

In your question, it asks how many license plate combinations we could make WITHOUT repeats.

We need some prior knowledge to answer this question.

We know that:

With the information we know above, we can solve the question.

Since we CAN'T have repeats, we would be excluding a letter or number for each license plate.

We're going to need to multiply each "section" in order to find how many combinations of license plates we can make.

We decrease by one letter and one number in each section since we can't have repeats.

Now, we can solve.

Work:

[tex]26*25*24*23*10*9 = 32,292,000[/tex]

When you're done multiplying, you should get 32,292,000.

This means that there could be 32,292,000 different combinations of license plates.

[tex]10\cdot9\cdot26\cdot25\cdot24\cdot23=32292000[/tex]

Answer:

(-inf,2]

Step-by-step explanation:

[tex](w \circ r)(x)=w(r(x))[/tex]

[tex]w(2-x^2)[/tex] I replaced r(x) with 2-x^2

[tex](2-x^2)-x[/tex] I replace the x in w(x)=x-2 with 2-x^2

[tex]-x^2-x+2[/tex]

You can graph this to find the range.

But since this is a quadratic (the graph is a parabola), I'm going to find the vertex to help me to determine the range.

The vertex is at x=-b/(2a).  Once I find x, I can find the y that corresponds to it by using y=-x^2-x+2.

Comparing ax^2+bx+c to -x^2-x+2 tells me a=-1, b=-1, and c=2.

So the vertex is at x=1/(2*-1)=-1/2.

To find the y-coordinate that corresponds to that I will not plug in -1/2 in place of x into -x^2-x+2.

This gives me

-(-1/2)^2-(-1/2)+2

-1/4 +  1/2  +2

Find a common denominator which is 4.

-1/4 +  2/4  +8/4

8/4

2.

So the highest y value is 2 ( I know tha parabola is upside down because a=negative number)

That mean then range is 2 or less than 2.

So the answer an interval notation is (-inf,2]

Answer:

the IQR is 41 because 62-21=41

Answer:

D

Step-by-step explanation:

The rules we need to simplify this are:

[tex]a^x*a^y=a^{x+y}[/tex]

and

[tex](a^x)^y=a^{xy}[/tex]

Now, let's simplify the problem:

[tex](y^{-7}*y^{-3})^{-1}\\(y^{-10})^{-1}\\y^{10}[/tex]

Answer choice D is y^10, so that's correct.

Answer:  

t2=ar^(2-1)

20=ar

then

t4=ar^(4-2)

45/4=ar.r

45/4=20.r

45/80=r

Answer:

r=±0.75

Step-by-step explanation:

Given:

a2= 20

a4= 45/4

As a geometric sequence has a common ratio and is given by:

an=a1(r)^n-1

where

an=nth term

a1=first term

n=number of term

r=common ratio

Now

a2=20=a1(r)^(2-1)

20=a1(r)^1

20=a1*r

Also

a4=45/4=a1(r)^(4-1)

45/4=a1r^3

(a1*r)r^2=45/4

Substituting value of 20=a1*r

(20)r^2=45/4

r^2=45/4(20)

r^2=0.5625

r=±0.75!

Answer:

A C D

Step-by-step explanation:

A prime Trinomial is one that cannot be reduced into integer factors. It is easier to define a non prime trinomial first.

x^2 - 4x - 12 can be factored into

(x - 6)(x + 2) -6 and 2 are integers.

So B is not prime.

All of the others are prime. You need to use the quadratic formula to solve them.

B factors into

x^2 - 13x + 42

(x - 7)(x - 6)

The value of r that satisfies the given area is approximately 5.08 ft, which is closest to 4 ft.

The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius. In this case, the area of the circle is given as 64 ft². So we can substitute the value of A as 64 ft² in the formula to find the value of r.

64 = πr²

To solve for r, we can divide both sides of the equation by π and then take the square root of both sides.

r² = 64/π

r = √(64/π)

To simplify further, we can approximate the value of π as 3.14.

r = √(64/3.14)

r ≈ 5.08 ft

Based on the given options, the value of r that is closest to 5.08 ft is 4 ft.

Answer:

ITS 8FT

Step-by-step explanation:

B). 1/2

In this question, it's asking to find the probability of Ethan getting a side of the cube that is less than 4.

In this case, we know that Ethan is rolling a 6 sided number cube, meaning that the numbers on the cube will range from 1-6.

On the cube, we need to get the numbers that are less than 4.

1

2        ← These numbers are less than 4.

3  

___

4

5

6

Knowing how many numbers are less than 4, we can solve the question.

We know that there are 3 numbers less than 4. So that will be our numerator.

There are 6 numbers in total, so that will be our denominator.

We can represent probability as a fraction.

Your fraction should look like this:

[tex]\frac{3}{6}[/tex]

We are not done yet, we would need to simplify the fraction.

To simplify, we would just divide the numerator and denominator by 3.

[tex]\frac{3}{6} \div \frac{3}{3}=\frac{1}{2}[/tex]

Once you're done solving, you should get [tex]\frac{1}{2}[/tex]

This means that answer choice B). 1/2 would be the correct answer.

The probability that Ethan gets a number less than 4 by rolling the dice is 1/2.

The probability of an event is the chance of happening that particular event.

The number of events in the sample space when rolling a dice is = 6.

The number of favorable events in that sample space = Getting a number less than 4 = 3.

Therefore, the probability of this particular event is

= 3/6

= 1/2

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Arc length will be equal to the perimeter divided by 4 since we are talking about quarter circle.

[tex]2\pi r/4=\pi r/2=6\pi/2\approx\boxed{9.42}[/tex]

Hope this helps.

r3t40

Answer:

9.42 inches

Step-by-step explanation:

the cercle length is : 2π×r = 2×3.14 ×6=37.68 inches

the arc length of a quarter circle is : 37.68/4 =9.42 inches

Answer:

She had eaten 15 chocolates.

Step-by-step explanation:

[tex]\frac{5}{8} * 24 = \frac{120}{8} = 15[/tex]

Final answer:

Kelsey ate 15 chocolates out of the 24 after a week, which is 5/8 of the total. Jenny initially had 14 chocolates before eating two and giving half of the remainder to Lisa.

Explanation:

To solve how many chocolates Kelsey has eaten, we have to calculate 5/8 of 24. Since 24 chocolates multiplied by 5/8 equals 15, Kelsey ate 15 chocolates after one week.

Regarding the second question about Jenny and her chocolates:

In summary, the answer to how many chocolates Jenny initially had is 14.

Answer:

[tex]\large\boxed{h^{-1}(x)=\dfrac{3x+4}{2}}[/tex]

Step-by-step explanation:

[tex]h(x)=\dfrac{2x-4}{3}\to y=\dfrac{2x-4}{3}\\\\\text{Exchange x to y and vice versa}:\\\\x=\dfrac{2y-4}{3}\\\\\text{Solve for y:}\\\\\dfrac{2y-4}{3}=x\qquad\text{multiply both sides by 3}\\\\2y-4=3x\qquad\text{add 4 to both sides}\\\\2y=3x+4\qquad\text{divide both sides by 2}\\\\y=\dfrac{3x+4}{2}[/tex]

Answer:

a) The length of an arc between  two consecutive fan blades is 16.76 inches

b) The area of each sector is 67.02 inches²

c) The angular velocity is 4m rad/sec

Step-by-step explanation:

* Lets explain how to solve the problem

- The fan has three thin blades that spin to  produce a breeze

- The diameter of the fan is  16 inches

- The three blades divided the circle into three equal parts

- The circumference of the circle is 2πr

a)

∵ The diameter of the circle = 16 inches

∵ The radius of the circle is half the diameter

∴ The radius (r) = 1/2 × 16 = 8 inches

∵ The length of the circle = 2πr

∴ The length of the circle = 2π(8) = 16π

- The length of an arc between  two consecutive fan blades is 1/3

  the length of the circle

∴ The length of the arc = 1/3 × 16π = 16.76 inches

* The length of an arc between  two consecutive fan blades is

  16.76 inches

b)

- The area of a sector in the circle = [tex]\frac{x}{360}(\pi r^{2})[/tex]

  where x is the central angle of the sector and r is the radius

  of the circle

∵ The angle between each two consecutive blades = 360°/3

∴ x = 360°/3 = 120°

∵ r = 8 inches

∴ The area of each sector = [tex]\frac{120}{360}(\pi )(8^{2})=67.02[/tex]

* The area of each sector is 67.02 inches²

c)

∵ The angular velocity = Ф rad ÷ t, where Ф is the central angle

   with radian measure and t is the time in seconds

∴ ω = Ф/t  radian/second

∵ Ф = 8m radians

∵ t = 2 seconds

∴ ω = 8m ÷ 2 = 4m rad/sec

* The angular velocity is 4m rad/sec

Answer:

Step-by-step explanation:

[tex]C(F)=\dfrac{5}{9}(F-32)[/tex]

The function C(F) represents the temperature of F degrees Fahrenheit converted to degrees Celsius.

The function C(F) represents the temperature of F degrees Fahrenheit converted to degrees Celsius. Temperature conversion is the process of converting between different temperature scales, such as Fahrenheit (°F), Celsius (°C), and Kelvin (K).

Each scale measures temperature differently, so conversions are necessary for various applications. The most common conversion formulas are: Celsius to Fahrenheit: °F = (°C × 9/5) + 32, and Celsius to Kelvin: K = °C + 273.15. These conversions are vital for international communication, weather reports, and scientific calculations.

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Answer:

See below in bold.

Step-by-step explanation:

The 50 grams of alloy contains 50 * 0.52 = 26 g of pure copper.

So the amount of copper n the mixture = 78 + 26 = 104 g.

The total mass of the mixture = 78 + 50 = 128 g.

% copper = 104 * 100 / 128

= 81.25%.

Final answer:

The resulting mixture contains 104 grams of copper, which constitutes 81.25% of the mixture.

Explanation:

To find out how many grams of copper are in the resulting mixture of the alloy, we need to calculate the amount of copper from both sources and add them together. The first source is 50 grams of an alloy that is 52% copper, and the second source is 78 grams of pure copper.

Now, add the amounts of copper from both sources:

26 grams (from the alloy) + 78 grams (pure copper) = 104 grams of copper.

To determine what percent of the resulting mixture is copper, we add the total weight of the mixture (50 grams + 78 grams = 128 grams) and then calculate the percentage:

(104 grams copper / 128 grams total) × 100 = 81.25%

So, the resulting mixture contains 104 grams of copper and is 81.25% copper.

[tex]\bf \begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array}~\hfill \begin{array}{llll} \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 2\log_2(x)+\log_2(x+5)\implies \log_2(x^2)+\log_2(x+5) \\\\\\ \log_2[x^2(x+5)]\implies \log_2(x^3+5x^2)[/tex]

The expression 2log2x + log2(x + 5) can be simplified to a single term, log2[x² × (x + 5)], by using properties of logarithms.

The subject of this question is the simplification of a logarithmic expression. In the mentioned expression, we have 2log2x + log2(x + 5), and we'll use the properties of logarithms to simplify it.

First remember that in terms of logarithms, mlogₐ(b) = log_a(b^m). So we can take the coefficient of the first term and use it as an exponent, which transforms '2log2x' into 'log2(x²)'.

Next, note the law that states logₐ(b) + log(c) ₐ= logₐ(bc). This means that we can combine the two terms under a singular logarithm to give us a single term: log2[x² × (x + 5)]. So, the simplified form of the expression 2log2x + log2(x + 5) is log2[x² × (x + 5)].

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Answer:

The second polynomial is:

[tex]6d^5-2c^3d^2+5c^2d^3-12cd^4+8[/tex]

Step-by-step explanation:

Given

[tex]Sum\ of\ polynomials=S=8d^5-3c^3d^2+5c^2d^3-4cd^4+9\\Polynomial\ 1=A=2d^5-c^3d^2+8cd^4+1\\Polynomial\ 2=B=?\\S=A+B\\B=S-A\\=8d^5-3c^3d^2+5c^2d^3-4cd^4+9-(2d^5-c^3d^2+8cd^4+1)\\=8d^5-3c^3d^2+5c^2d^3-4cd^4+9-2d^5+c^3d^2-8cd^4-1\\=8d^5-2d^5-3c^3d^2+c^3d^2+5c^2d^3-4cd^4-8cd^4+9-1\\=6d^5-2c^3d^2+5c^2d^3-12cd^4+8[/tex]

Answer:

5/16.

Step-by-step explanation:

The probability of getting a correct answer on one question is 1/2 and the probability of getting it wrong is  1 - 1/2 = 1/2.

The probability of the first 2 being correct and the next 3 wrong = (1/2)^5 = 1/32.

There are  10  possible combinations of this ( first wrong the 2nd and third right and last  2 wrong,  etc).

So the  required probability is 10 * 1 /32

= 10/32

= 5/16.

Answer:

[tex]Pr=\dfrac{45}{208}\approx 0.216.[/tex]

Step-by-step explanation:

You are given the information about students:

In total, there are 45 + 60 + 82 + 21 = 208 students.

Use the definition of the probability:

[tex]Pr=\dfrac{\text{Number of favorable outcomes}}{\text{Number of all possible outcomes}}[/tex]

Number of favorable outcomes = 45

Nomber of all possible outcomes = 208,

hence,

[tex]Pr=\dfrac{45}{208}\approx 0.216.[/tex]

Add all of the numbers together

2+3.5+2+3.5

5.5+5.5

= 11 cm

Answer is 11cm - second time

Answer:

11 cm

Step-by-step explanation:

The perimeter is equal to

P =2(l+w)

P = 2(2+3.5)

  = 2(5.5)

  = 11 cm

Answer:

22

Step-by-step explanation:

11+22+33=66

66/3

=22

Answer:

22

Step-by-step explanation:

The mean is just another word for average

Add up the three numbers and divide by 3

(11+22+33) /3 = 66/3 = 22

Answer:

A.  True.

Step-by-step explanation:

The value increases by the same amount ($5) every year so it is simple interest.

Answer:True

Step-by-step explanation: