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2. A cube-shaped aquarium has edges that are 3 ft long. The aquarium is filled with water that has a density of 62 lb/ft^3
(a) Should the aquarium be placed on a table that can support a maximum weight of 200 lb? Explain why or why not.
(b) Would the density of the water change if the aquarium was only half full? Explain.

We have been given that L'shanda can choose between 3 sweaters and 4 skirts.

We need to figure out the number of ways in which L'shanda can pick 1 sweater and 1 skirt out of the available items.

We need to use combinations here. First of all, we will figure out the number of ways of choosing 1 sweater out of 3 available sweaters. And then we will determine the number of ways of choosing 1 skirt out of 4 available skirts.

Number of ways of choosing 1 sweater = [tex]_{1}^{3}\textrm{C}=\frac{3!}{2!1!}=\frac{3\cdot 2!}{2!} = 3[/tex]

Number of ways of choosing 1 skirt = [tex]_{1}^{4}\textrm{C}=\frac{4!}{3!1!}=\frac{4\cdot 3!}{3!} = 4[/tex]

Therefore, total number of ways to select 1 sweater and 1 skirt are [tex]3\cdot 4 = 12[/tex]

Answer:

0.51

Step-by-step explanation:

10^1=divide by 10

5.1/10=0.51

The area of the rectangle will be 384 ft². Square feet and other similar units are used to measure area.

The space filled by a flat form or the surface of an item is known as the area.

The number of unit squares that cover the surface of a closed-form is the figure's area.

Let the length of a rectangle is,l and the width of a rectangle be,w

Given condition;

l=6w

The perimeter of the rectangle  = 112 ft,

⇒2l+2w=112

According to condition;

2(6w)+2w=112

12w+2w=112

14w=112

w=8

The length of a rectangle is;

l=6w

l=48

Area of rectangle = length × width

A =lw

A=6 × 48 ft

A=384 ft^2

Hence the area of the rectangle will be 384 ft².

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The calculated value of the probability of 4

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

The spinner

Where, we have

Sections = 8

Also, we have

Sections that read 4 = 1

Using the above as a guide, we have the following:

P(4) = 1/8

Hence, the probability of 4 is 1/8

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Answer:

N55.0679°W

Step-by-step explanation:

Check the attachment for detailed explanation.

Final answer:

Using trigonometry and the sine function, the maximum height that the ladder can reach when raised to a 75-degree angle with a length of 60 feet is approximately 58 feet.

Explanation:

To determine the maximum height that a fire truck ladder can safely reach when it is extended to 60 feet in length and raised to a maximum angle of 75 degrees with the horizontal, we would use trigonometry, specifically the sine function. The sine of an angle in a right triangle is equal to the opposite side (height in this case) divided by the hypotenuse (the length of the ladder).
Therefore, the maximum height (h) can be calculated using the formula:

h = Ladder Length × sin(Angle)

Where:

Ladder Length is 60 feet

Angle is 75 degrees

Plugging in our values we get:

h = 60 feet × sin(75 degrees)

Using a calculator, we find that:

sin(75 degrees) is approximately 0.9659.

Therefore:

h ≈ 60 feet × 0.9659

h ≈ 57.95 feet

The maximum height the ladder can safely reach when disregarding the height of the fire truck itself is approximately 58 feet.

We have been given a polynomial [tex]f(x)=2x^{3} +x^{2}-3x+1[/tex] and we are asked to find average rate of  change from x = −1 to x = 1.

First of all we will find f(-1) and f(1).

[tex]f(-1)=2\cdot (-1)^{3} +(-1)^{2}-3(-1)+1[/tex]

[tex]f(-1)=2\cdot (-1) +1+3+1[/tex]

[tex]f(-1)=-2 +5=3[/tex]

Let us find f(1),

[tex]f(1)=2\cdot (1)^{3} +(1)^{2}-3(1)+1[/tex]

[tex]f(1)=2\cdot 1 +1-3+1[/tex]

[tex]f(1)=2+1-3+1[/tex]

[tex]f(1)=4-3=1[/tex]

Now let us find slope for our values.

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

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

[tex]m=\frac{1-3}{1--1}[/tex]    

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

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

Therefore, average rate of change from our given x values will be -1.

The equation of the image circle is,

(x + 4)² + (y + 5)² = 9.

The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

Since, We know that;

After reflect a point over the line y = -1, we need to use the formula ;

(x,y) → (x, -y - 2).

Hence, We can apply this formula to each point on the circle and simplify to get the equation of the image circle:

(x + 4)² + (y - 3)²= 9

(x + 4)² + (-y - 2 - 3)² = 9

(x + 4)² + (y + 5)² = 9

So, the equation of the image circle is,

(x + 4)² + (y + 5)² = 9.

Therefore, the correct option is,

(C) (x + 4)² + (y + 5)² = 9.

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the probability of getting a number greater than or equal to 2 when rolling a fair number cube is [tex]\( \frac{5}{6} \).[/tex]

When rolling a fair number cube (also known as a fair six-sided die), each face has an equal probability of showing up. Since there are 6 faces numbered 1 to 6, the probability of getting any specific number (including 2) on a single roll is [tex]\( \frac{1}{6} \)[/tex], assuming the die is fair.

To find the probability of getting a number greater than or equal to 2, we can count the favorable outcomes (rolling a number 2, 3, 4, 5, or 6) and divide by the total possible outcomes (which is 6 for a fair six-sided die).

Favorable outcomes: 2, 3, 4, 5, 6 (total of 5 numbers)

Total possible outcomes: 6 numbers

So, the probability of getting a number greater than or equal to 2 when rolling a fair number cube is [tex]\( \frac{5}{6} \).[/tex]

Answer with explanation:

 When we add, subtract or multiply or in some cases division is done between two or more Polynomials then we can get new polynomials.

This can be explained in following way

   [tex]1.\rightarrow (x^3+x^2+3x+4) +(x^4+5x+6)\\\\=x^4+x^3+x^2+8x+10\\\\2.\rightarrow (x^3+x^2+3x+4) -(x^4+5x+6)\\\\=-x^4+x^3+x^2-2x-2\\\\3.\rightarrow (x+2)\times x^2\\\\=x^3+2x^2\\\\4.\rightarrow \frac{x^3+2x^2}{x}\\\\\rightarrow x^2+2x[/tex]

Real world Situation

There is a large enclosed room .We want to place bulbs in ceiling.To do that,we draw few straight lines that is Linear polynomial columnwise and then we have drawn Linear polynomial Row wise.The point of intersection of these lines gives the points where the bulbs should be fixed.Now ,if we join these points where the bulb is placed we will get a new polynomial.

The equation of the circle is given by (x-h)^2 + (y-k)^2 = r^2 where (h,k) represent the center of the circle and r represents the radius

From the question , we have been given (h,k) = (2,-5) and r =12

(x-2)^2 + (y-(-5))^2 = 12^2

The two minuses in the second term cancel out and give a plus. (-*- = +)

(x-2)^2 + (y+5)^2 = 144

So, the equation of the circle is:

(x-2)^2 + (y+5)^2 = 144 (Option A)

The answer is (x + 3) 2 + (y − 1) 2 = 81

To determine whether quantities vary directly or inversely, let's first define the variables:

Let x be which student essays had been graded.

Let y be the time taken to grade those essays.

In this case, it seems that the time taken to grade essays varies inversely with the number of essays graded. This means that as the number of essays increases, the time taken to grade them decreases, assuming the grading speed remains constant.

So, we can set up an equation using the constant of variation: xy = k

where the variation in the ratio is designated by k.

Given that the teacher grades 25 essays in 4 hours, we can use this information to find the constant of variation:

25 × 4 = k

k = 100

Now that we have the constant of variation, we can use it to find the time it takes to grade 35 essays:

35y = 100

y = (100/35)

y ≈ 2.857

So, it will take approximately 2.857 hours to grade 35 essays.

To determine if quantities vary directly or inversely, we compare how changes in one quantity affect the other.

Here, as the number of essays increases, the time taken to grade them decreases, indicating an inverse relationship.

Using the formula for inverse variation, xy = k, where x is the number of essays and y is the time taken, we find the constant of variation by substituting values: 25 × 4 = k, which gives k=100.

With the constant, we can find the time to grade 35 essays: 35y = 100.

Solving for y, we get y ≈ 2.857 hours. Therefore, it will take approximately 2.857 hours to grade 35 essays.

Complete Question:

Determine whether quantities vary directly or inversely and find the constant of variation. A teacher grades 25 students' essays in 4 hours. Assuming he grades at the same speed, how long will it take him to grade 35 essays?