please help asap 45 pts

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?

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.

The permutation is defined as the arrangement number or order in which rearrangement of  element in an order list.

(a) The letters in the word computer be arranged in a row is 40320.

(b) The letters of the word computer be arranged 5040 ways.

Given:

The given latter is COMPUTER.

(a)

There are 8 letter in word COMPUTER. Calculate the number of ways in which we can arrange 8 letters.

[tex]8!=8\times7\times6\times5\times4\times3\times2\times1\\8!=40,320.[/tex]

Thus, the letters in the word computer be arranged in a row is

(b)

As per the question if we treat CO as a unit then there will only seven letters effective.

[tex]7!=7\times6\times5\times4\times3\times2\times1\\7!=5,040[/tex]

Thus, the letters of the word computer be arranged 5040 ways.

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Answer:

1536

Step-by-step explanation:

Given : A geometric sequence 0.75, 1.5, 3, 6, . . .

To Find:Find the 12th term

Solution:

0.75, 1.5, 3, 6, . . .

First term =a = 0.75

Common difference = [tex]r = \frac{a_3}{a_2} =\frac{a_2}{a_1}=\frac{1.5}{0.75}=\frac{3}{1.5}=2[/tex]

nth term of G.P. = [tex]a_n=a r^{n-1}[/tex]

Where a = First term = 0.75

r = common difference = 2

Substitute n = 12

[tex]a_{12}= 0.75 (2)^{12-1}[/tex]

[tex]a_{12}= 0.75 (2)^{11}[/tex]

[tex]a_{12}= 1536[/tex]

Hence the 12th term in the geometric sequence. 0.75, 1.5, 3, 6, . . . is 1536.

Answer:

Step-by-step explanation:

45

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

The graphs match the equations as follows:

Graph A: (−x+3)^2 -2

Graph B: (x−2)^2+3

Graph C : −(x+2)^2 +3

Graph D : 2(x−2)^2 +3

Equation (−x+3)^2 -2:

This equation is in the form of a quadratic function, y=a(x−h) ^2+k, where (h ,k) is the vertex of the parabola.

The vertex of the parabola is at (3,1), which matches the vertex of Graph A.

The coefficient a is negative, which tells us that the parabola opens downward.

The constant term k is −2, which tells us that the minimum value of the parabola is −2.

Equation (x−2)^2+3 :

This equation is in the form of a quadratic function, y=a(x−h) ^2 +k, where (h, k) is the vertex of the parabola.

The vertex of the parabola is at (2,3), which matches the vertex of Graph B. The coefficient a is positive, which tells us that the parabola opens upward.

The constant term k is 3, which tells us that the minimum value of the parabola is 3.

Equation  −(x+2)^2 +3:

This equation is in the form of a quadratic function, y=a(x−h) ^2 +k, where (h, k) is the vertex of the parabola.

The vertex of the parabola is at (−2,3), which matches the vertex of Graph C.The coefficient a is negative, which tells us that the parabola opens downward.

The constant term k is 3, which tells us that the maximum value of the parabola is 3.

Equation  2(x−2)^2 +3 :

This equation is in the form of a quadratic function, y=a(x−h) ^2 +k, where (h ,k) is the vertex of the parabola.

The vertex of the parabola is at (2,3), which matches the vertex of Graph D. The coefficient a is positive and multiplied by 2, which tells us that the parabola opens upward and is stretched vertically by a factor of 2.

The constant term k is 3, which tells us that the minimum value of the parabola is 3.

To determine the equation of the line with x-intercept 8 and y-intercept -2, the slope is calculated to be 1/4. Thus, the line's equation in slope-intercept form is y = (1/4)x - 2.

To find the equation of a line with an x-intercept of 8 and a y-intercept of -2, we'll first determine the slope of the line (m) and then use the slope-intercept form (y = mx + b).

Two points on this line are (8,0) and (0,-2). The slope (m) of a line is calculated by the change in y divided by the change in x, which in this case is
(0 - (-2)) / (8 - 0) = 2 / 8 = 1 / 4.

Now we know the slope is 1/4 and the y-intercept (b) is -2, so the equation of the line in slope-intercept form is:

y = (1/4)x - 2

Answer:

D

Step-by-step explanation:

Answer with explanation:

It is given that, a plane intersects a double-napped cone such that the plane intersects both Nappes through the cone's vertex.It also depends on how thick or thickness of the plane:

1.We can surely say that degenerate hyperbola will be formed,whatever be the thickness of Plane.

2. Two parallel lines=pair of intersecting lines ,because both ,A double napped cone and Plane is three dimensional shape.Instead of intersecting lines ,two parallel lines which we call pair of intersecting lines,  will be formed.

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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[tex] |\Omega|=10\\
|A|=1\\\\
P(A)=\dfrac{1}{10}=10\% [/tex]

First arrange the words in order:

AAABNN

so there are 3 A's, 1 B and 2 N's.

If there were no constraints,

Number of permutations = (3+1+2)!/(3!1!2!)=6!/(6*1*2)=720/(6*2)=60

Number of words that begin with B is to take out the B, that leaves us with

AAANN => 3 A's and 2 N's

=> 5!/(3!2!) = 120/(6*2)=10

Therefore the number of words that do NOT begin with B is

60-10 = 50 ways

Final answer:

The area of a right triangle with sides of 5 inches and 12 inches as the base and height, respectively, is 30 square inches, calculated using the formula A = (base × height) / 2.

Explanation:

The area of a right triangle is calculated by multiplying one-half the base by the height: A = (base × height) / 2.

In the case of a right triangle with sides measuring 5 inches, 12 inches, and 13 inches, the 5-inch and the 12-inch sides are the legs and thus can be used as the base and height. Using the formula for the area of a right triangle:

A = (5 × 12) / 2

A = 60 / 2

A = 30 square inches

Therefore, the area of this triangle is 30 square inches.

The probability of getting someone who carpools or someone who works full time is 0.36.

To find the probability of getting someone who carpools or someone who works full time, we need to add the probabilities of the two events.

The probability of getting someone who carpools is 6/100 and the probability of getting someone who works full time is 30/100.

So the probability of either event happening is (6/100) + (30/100) = 36/100 = 0.36.

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