50 POINTS!
Classify the following angle.
Show your work.

the minimum surface area for the box is 4800 cm².

Forming the Equation of SurfaceArea

It is given that the given rectangular box is square-based and top is open. hence, it consists of  square and 4 rectangles.

Let the side of the square be a, and height of the box be h.

Then, the total surface area of the box will be given by,

S = a² + 4ah _________ (1)

Also, it is given that the volume of the box is, V = 32000 cm³

The volume of the rectangular box, V = a² h

Eliminating One of the Variables From the Equation

The volume of the rectangular box, V = a² h

⇒ a² h = 32000

⇒ h = 32000/a² _______ (2)

Substituting this value of h in equation (1), we get,

S = a² +4a(32000/a²)

S = a²+128000/a

Minimizing the Surface Area Equation

To find the minima, put, dS/da = 0

dS/da = 2a-128000/a²

⇒ 2a-128000/a² = 0

Multiplying the whole equation with a², we get,

2a³-128000 = 0

⇒ 2a³ = 128000

⇒ a³ = 128000/2

⇒ a³ = 64000

⇒ a = 40 cm

Calculating the Minimum Surface Area

From, equation (2), h = 32000/(40)²

h = 32000/1600

h = 20 cm

Now, substituting the computed values of a and h in equation (1), we get,

S = (40)² +4(40)(20)

S = 1600 +3200

S = 4800 cm²

∴ The minimum surface area of the box is 4800 cm².

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The two consecutive integers are 5 and 6

let

the first integer = x

second integer = x + 1

Therefore,

x(x + 1) = 30

x² + x = 30

x² + x - 30 = 0

(x - 5)(x + 6) = 0

Therefore,

x = 5 and  x = -6

Hence, let's use 5 because it's positive.

So, the numbers are 5 and 6

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Answer: (4,-7)

Step-by-step explanation:

when f(x) = [tex]x^3[/tex] the vertex is (0,0). when g(x)= [tex](x-4)^3-7[/tex] the vertex will be (4,-7). The equation is put into vertex form, in order for us to find the vertex (h,k), followed by [tex]y=a(x-h)^3+k[/tex]. With h=4 and k=-7.

Therefore, the answer is (4,-7)

Variation is a topic that describes the relationship between or among given variables. So that the expected answers to the questions are;

Variation is a topic that can be used to show the relationship between or among a given number of variables. The types are direct variation, inverse variation, constant variation, and joint variation.

The required answers to the given questions are:

1. y [tex]\alpha[/tex] x

 ⇒ y = kx

where k is the constant of proportionality.

Given that, y = 5 and x =3, then;

5 = 3x

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

Thus,

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

2. v [tex]\alpha[/tex] w

  v = kw

Given: v = 21, w = 8

Then,

21 = 8k

k = [tex]\frac{21}{8}[/tex]

Thus,

v = [tex]\frac{21}{8}[/tex]w

3.  q [tex]\alpha[/tex] [tex]\frac{1}{q}[/tex]

  y = [tex]\frac{k}{q}[/tex]

Given: q = 5, q = 6

Then,

5 = [tex]\frac{k}{6}[/tex]

k = 5 x 6

k = 30

Thus,

p = [tex]\frac{30}{q}[/tex]

4. y [tex]\alpha[/tex] [tex]\frac{1}{x}[/tex]

  y = [tex]\frac{k}{x}[/tex]

Given: y = 11, x = 3

Then,

11 = [tex]\frac{k}{3}[/tex]

k = 11 x 3

k = 33

Thus,

y = [tex]\frac{33}{x}[/tex]

5. F [tex]\alpha[/tex] [tex]\frac{1}{l}[/tex]

F = [tex]\frac{k}{l}[/tex]

Given; F = 20 pounds, l = 1.5 ft

20 = [tex]\frac{k}{1.5}[/tex]

k = 20 x 1.5

k = 30

F = [tex]\frac{30}{l}[/tex]

When l = 6 ft, then;

F = [tex]\frac{30}{6}[/tex]

F  = 5 pounds

The amount of force required is 5 pounds.

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The angle that the ramp makes with the ground is 16 degrees.

We can see that the ramp is and its support now constitutes a right angled triangle.

Thus, the hypotenuse of the triangle is 40feet while the opposite of the angle is 11 feet.

The angle made with the ground is;

sin x = 11/40

x = sin-1(11/40)

x = 16 degrees

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The situation in which the quotient of -24 divided by 3 be used to answer the question is;

Check all options:

A. The temperature of a substance decreased by 24∘C per minute for 3 minutes. What was the overall change of the temperature of the substance?

Temperature decrease = 24°C per minutes × 3 minutes

= 72°C decrease in temperature

B. A football team lost 24 yards on one play, then gained 3 yards on the next play. How many total yards did the team gain on the two plays?

Loss = -24 yards

Gain = 3 yards

Total gain = -24 + 3

= -21 yards

C. Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?

Amount withdrawn per day = -$24 / 3

= -$8 per day

Change in her account per day = $8

D. A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?

Number of children = 24 cookies / 3 cookies

= 8 cookies per children

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Answer:

(4,-3)

Step-by-step explanation:

Multiplying the bottom equation by -6, we get -6x+12y=-60.

Adding this to the first equation gives 15y=-45, and thus y=-3.

So, it follows that 6x+3(-3)=15, and thus x=4.

There is not enough evidence at the 0.02 level that the doors are either too long or too short. So, the null hypothesis is accepted.

Using the p-value approach for the hypothesis test, whether the null hypothesis is rejected or accepted.

It is given that,

Sample size n = 17

Sample mean X = 2069

Standard deviation σ = 24

Population mean  μ = 2058

Significance level α = 0.02

Hypothesis:

The null hypothesis: H0: μ = 2058

The alternative hypothesis: Ha: μ ≠ 2058

So, the z-test score for the given distribution is,

z-score = (X - μ)/(σ/√n)

On substituting,

z-score = (2069 - 2058)/(24/√17)

            = 11/5.82

            = 1.89

Thus, the p-value for the z-score of 1.89 is 0.0587 ≅ 0.06.

Since 0.06 > 0.02(significance level), the null hypothesis is accepted. So, there is no sufficient evidence that the doors are either too long or too short.

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Answer: 15 square units

Step-by-step explanation:

[tex]A=(5)(3)=15[/tex]

Answer:

  II only

Step-by-step explanation:

The positive leading coefficient and even degree of f(x) tell you its graph is U-shaped and opens upward.

If there is an interval where the function is negative, it must lie between the two x-intercepts. The graph will be above the x-axis (f(x) > 0) for x-values outside that interval.

The only reasonable answer choice is ...

  II  (-1, 4) . . . . only

Answer:

y=-4x+21

Step-by-step explanation:

1. Draw these points on a graph.

2. Point A is (4,5) and Point B is (7,-7).

3. You find the slope (gradient) by the Change in Height/ Change in Horizontal distance.

4. Now put that slope and one point into the "Point-Slope Formula"

5. Simplify Algebraically.

Answer:

y = - 4x + 21

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (4, 5 ) and (x₂, y₂ ) = (7, - 7 )

m = [tex]\frac{-7-5}{7-4}[/tex] = [tex]\frac{-12}{3}[/tex] = - 4 , then

y = - 4x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, 5 )

5 = - 16 + c ⇒ c = 5 + 16 = 21

y = - 4x + 21 ← equation of line

Answer:

$7128.80

Step-by-step explanation:

First, create an equation.

Use the formula f(t)=P(1+b)^t

Plug in the information.

Time will equal months.

f(12)=5000(1+0.03)^12

Solve.

f(12) = 7128.80

Myron was left with $7128.80.

Hope this helps!

Answer:

The equation is 21x+4y<3150

Step-by-step explanation:

This question can be understood by the concept of inequalities.

Let x be the number of stocks of software company and y be the number of stocks of robotics company purchased.

Money invested in 1 stock of software company = $42

Money invested in 1 stock of robotics company = $8

Total expenditure on software company stock = 42*x

Total expenditure on robotics company stock = 8*y

So the total expenditure should be less than 6300

Hence the equation is

42x+8y<6300

21x+4y<3150

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The probability that the next toss will be heads is 1/8.

The likelihood of an event occurring is described by probability. We frequently have to make forecasts about the future in real life. We may or may not be aware of the outcome of an event. When this happens, we declare that there is a chance the event will take place.

Using the probability formula, one can determine the likelihood of an event by dividing the favorable number of possibilities by the total number of options. Since the favorable number of outcomes can never be greater than the entire number of outcomes, the probability of an event happening can range from 0 to 1.

Probability of getting two tails and next heads in three tosses is,

=1/2*1/2*1/2

=1/8

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The calculated value of the total area of the figure is 49 square units

From the question, we have the following parameters that can be used in our computation:

The figure

Where, we have

Shape = trapezoid

The total area of the trapezoid is calculated as

Area = 1/2 * Sum of parallel sides * Height

So, we have

Area = 1/2 * (11 + 3) * 7

Evaluate

Area = 49

Hence. the total area of the figure is 49 square units

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keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the line above

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{-1}~,~\stackrel{y_2}{3}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{3}-\stackrel{y1}{(-2)}}}{\underset{run} {\underset{x_2}{-1}-\underset{x_1}{4}}} \implies \cfrac{3 +2}{-1 -4}\implies \cfrac{5}{-5}\implies \cfrac{1}{-1}\implies -1 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{\cfrac{1}{-1}} ~\hfill \stackrel{reciprocal}{\cfrac{-1}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{-1}{1}\implies 1}}[/tex]

Answer:

y =  x - 2

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (4, 2 ) and (x₂, y₂ ) = (0, - 2 )

m = [tex]\frac{-2-2}{0-4}[/tex] = [tex]\frac{-4}{-4}[/tex] = 1

the line crosses the y- axis at (0, - 2 ) ⇒ c = - 2

y = x - 2 ← equation of line

The probability that a prize ticket with a vowel will be chosen, not replaced, and then another prize ticket with a vowel will be chosen is 0.031 or 3.1%. So, this represents a dependent event.

It is given that there are 26 prize tickets in a bowl, labeled A to Z.

There are 5 vowels- A, E, I, O, U

So, the probability of choosing a ticket with a vowel will be - (5/26)

Another ticket with a vowel will be chosen without replacing the first ticket. Then, its probability is (4/25)

So, the required probability is,

(5/26) × (4/25) = 2/65 = 0.031

The event of choosing the first ticket with a vowel effect the event of choosing the second ticket with a vowel and vice-versa. So, this represents dependent events.

Therefore, the probability is 3.1 %.

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Answer:

  f(x) = (x -6)² +14

Step-by-step explanation:

Completing the square involves writing part of the function as a perfect square trinomial.

The square of a binomial results in a perfect square trinomial:

  (x -h)² = x² -2hx +h²

The constant term (h²) in this trinomial is the square of half the coefficient of the linear term: h² = ((-2h)/2)².

One way to "complete the square" is to add and subtract the constant necessary to make a perfect square trinomial from the variable terms.

Here, we recognize the coefficient of the linear term is -12, so the necessary constant is (-12/2)² = 36. Adding and subtracting this, we have ...

  f(x) = x² -12x +36 +50 -36

Rearranging into the desired form, this is ...

  f(x) = (x -6)² +14

__

Additional comment

Another way to achieve the same effect is to split the given constant into two parts, one of which is the constant necessary to complete the square.

  f(x) = x² -12x +(36 +14)

  f(x) = (x² -12x +36) +14

  f(x) = (x -6)² +14

The dimensions of the rectangle having greatest area in the piece of plexiglass which is a semicircle is [tex]2\sqrt{2}[/tex] and  [tex]\sqrt{2}[/tex].

Given radius of piece of plexiglass which is a semi circle equal to 2m.

We  have to find the dimensions of the rectangle which is inscribed in the semi circle.

Draw a rectangle in a semicircle.

Suppose the length of rectangle be x and width be y.

Draw a line from the center to the vertex of the rectangle which forms a right angled triangle with sides x,y and 2.

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

y=[tex]\sqrt{16-x^{2} }/2[/tex]

Area of rectangle=2xy

=x[tex]\sqrt{16-x^{2} }[/tex]/2

Finding the derivative of area

d A/dx=1/2[x(-2x)/2[tex]\sqrt{16-x^{2} }[/tex]+[tex]\sqrt{16-x^{2} }[/tex]]

d A/dx=0

1/2[[tex]-2x^{2} /2\sqrt{16-x^{2} }+\sqrt{16-x^{2} } ][/tex]=0

Solving for x

[tex]\sqrt{16-x^{2}[/tex]=[tex]x^{2} /\sqrt{16-x^{2} }[/tex]

16-[tex]x^{2}[/tex]=[tex]x^{2}[/tex]

16=2[tex]x^{2}[/tex]

[tex]x^{2}[/tex]=8

x=2[tex]\sqrt{2}[/tex]

put the value of x in y=[tex]\sqrt{16-x^{2} }/2[/tex] to get the value of y

y=[tex]\sqrt{16-8} /2[/tex]

=[tex]\sqrt{8} /2[/tex]

= 2[tex]\sqrt{2} /2[/tex]

=[tex]\sqrt{2}[/tex]

Hence the dimensions of rectangle which is in a semi circle of radius 2 is [tex]\sqrt{2} and 2\sqrt{2}[/tex].

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Answer:

Step-by-step explanation:

add 4 more rows like in the pattern and you will get your answer

Answer:

36 squares.

Step-by-step explanation:

Number of squares in each step is triangular number. This form a triangular series.

1 , 3 , 6 , 10........

Number of squares in each step is given by

  [tex]\sf \dfrac{n*(n+1)}{2}[/tex]  here , n is the step or term.

n = 8

[tex]\sf Number \ of \ squares \ in \ 8^{th} \ step = \dfrac{8 * (8+1)}{2}[/tex]

                                             [tex]\sf = \dfrac{8*9}{2}\\\\ = 4*9\\\\=36 \ squares[/tex]

The expression which is equivalent to 4x²√5x⁴ × 3√5x^8 is; 60x^8.

The expression given in the task content is; 4x²√5x⁴ × 3√5x^8.

Hence, upon evaluation of the product; we have;

= 12x²√25x¹².

= 60x^8.

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maybe try using the 2 and dropping the decimal

The equivalent expressions of T=A^1.5 are: T= (A)^3/2 and T= ((A)^1/2)^3

According to the statement

we have to show that the 1.5 is same as 3/2 are equivalent to T.

So,

Equivalent expressions are expressions with the same value.

The Given expression is

T=A^1.5

we can write it as a

T= (A)^3/2

Because we know that the

1.5 = 3/2

So, it become T= (A)^3/2.

And From

T=A^1.5

we can write it as a

T= ((A)^1/2)^3

because we know that the

3/2 = (1/2)^3.

So, The equivalent expressions of T=A^1.5 are: T= (A)^3/2 and T= ((A)^1/2)^3.

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On Monday: 35 mosquitos.

On Tuesday: 6 flies.

As you can notice in the diagram, the frog eats 3 flies for every 7 mosquitoes (for lunch). Then you can define this ratio as pursues:

3: 7 or 3/7.

Based on the table below:

If the frog swallows 15 flies on Monday, then the number of mosquitos that it eats can be estimated as pursues:

3/7 = 15/mosquitoes

mosquitoes [tex]$= \frac{15*7}{3}[/tex]

mosquitoes = 35

If the frog eats 14 mosquitoes on Tuesday, then the number of flies that it eats can be estimated as pursues:

3/7 = flies/14

flies [tex]$= \frac{14*3}{7}[/tex]

flies = 6

The complete question is:

A frog catches insects for his lunch. The frog likes to eat flies and mosquitoes in a certain ratio, which is shown in the diagram. The table shows the number of flies and the number of mosquitoes that the frog eats for two lunches.

Based on the ratio, complete the missing values in the table.

Day Flies Mosquitoes

Monday 151515

Tuesday

141414

Disregard Above

Use Picture

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Answer: A. (4, 4) and (4, 5)

Step-by-step explanation:

The same x value maps onto more than one y value.

Answer:

A)  

[tex]\begin{array}{|c|c|c|}\cline{1-3} t & H(t) & g(t)\\\cline{1-3} 1 & 60 & 20.4\\\cline{1-3} 2 & 76 & 30.8\\\cline{1-3} 3 & 60 & 41.2\\\cline{1-3} 4 & 12 & 51.6\\\cline{1-3} \end{array}[/tex]

Between 3 and 4 seconds.

B)  The baseballs will collide between 3 and 4 seconds.

Step-by-step explanation:

Given functions:

[tex]H(t)=-16t^2+64t+12[/tex]

[tex]g(t)=10+10.4t[/tex]

Substitute the values of t = 1, 2, 3 and 4 into the two functions:

[tex]H(1)=-16(1)^2+64(1)+12=60[/tex]

[tex]H(2)=-16(2)^2+64(2)+12=76[/tex]

[tex]H(3)=-16(3)^2+64(3)+12=60[/tex]

[tex]H(4)=-16(4)^2+64(4)+12=12[/tex]

[tex]g(1)=10+10.4(1)=20.4[/tex]

[tex]g(2)=10+10.4(2)=30.8[/tex]

[tex]g(3)=10+10.4(3)=41.2[/tex]

[tex]g(4)=10+10.4(4)=51.6[/tex]

Create a table with the found values:

[tex]\begin{array}{|c|c|c|}\cline{1-3} t & H(t) & g(t)\\\cline{1-3} 1 & 60 & 20.4\\\cline{1-3} 2 & 76 & 30.8\\\cline{1-3} 3 & 60 & 41.2\\\cline{1-3} 4 & 12 & 51.6\\\cline{1-3} \end{array}[/tex]

The solution to H(t) = g(t) is between 3 and 4 seconds as:

When t = 3, H(t) > g(t)

When t = 4, H(t) < g(t)

To prove this, equate the equations and solve for t:

[tex]\implies -16t^2+64t+12=10+10.4t[/tex]

[tex]\implies -16t^2+53.6t+2=0[/tex]

Using the Quadratic Formula to solve for t:

[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]

[tex]\implies t=\dfrac{-53.6 \pm \sqrt{53.6^2-4(-16)(2)} }{2(-16)}[/tex]

[tex]\implies t=\dfrac{53.6 \pm \sqrt{3000.96}}{32}[/tex]

[tex]\implies t=3.39, -0.04\:\:(\sf 2\:d.p.)[/tex]

As time is positive, t = 3.39 s (which is between 3 and 4 seconds).

When the two baseballs are at the same height they will collide.

Therefore, the baseballs will collide between 3 and 4 seconds (when t = 3.39 s).

A typical decision support system (DSS) includes many components including a(n) dialogue manager which allows decision makers to easily access and manipulate the DSS and to use common business terms and phrases.

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The length of diagonal is 9.5 ft

What is Pythagoras theorem?

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

We will find the length of diagonal as shown below:

l= 7 ft

h= 5 ft

w= 4 ft

Let diagonal be d.

Let v be the diagonal of the base.

Using Pythagoras theorem

[tex]v=\sqrt{7^2+4^2}[/tex]

[tex]=\sqrt{49+16}[/tex]

[tex]v=\sqrt{65}[/tex]

For finding the value of d again using Pythagoras theorem

[tex]d=\sqrt{v^2+h^2}[/tex]

[tex]=\sqrt{(\sqrt{65})^2+5^2 }[/tex]

[tex]=\sqrt{65+25}[/tex]

[tex]=\sqrt{90}[/tex]

=9.4868

Rounding to nearest tenth

=9.5

Hence, the length of diagonal is 9.5 ft

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The sequence that shows a pattern where each term is 1.5 times the previous term is option D

Series of numbers can be in a sequence either in arithmetic or in geometric. In the above question, it is geometric because the sequence shows a pattern where each term is 1.5 times the previous term.

Let us test each option one by one.

Option A

1.5 x -4 = -6

But the next number in the series is 6

Option B

1.5 x 10 = 15

1.5 x 15 = 22.5

But the next number in the series is 25

Option C

1.5 x 98 = 147

But the next number in the series is 99.5

Option D

1.5 x -200 = -300

1.5 x -300 = -450

1.5 x -450 = -675

Therefore, the sequence that shows a pattern where each term is 1.5 times the previous term is option D

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Answer: D (-200,-300,-450,-675,...)

Step-by-step explanation:

Recognizing the pattern in a sequence is used to write the function that represents it.