8th grade math
Geometry

Answer: To create a linear model that represents the amount of fuel on the plane as a function of flight time, we need to use the given information to calculate the fuel burn rate of the airplane in gallons per hour.

Fuel burn rate = Fuel capacity ÷ Range

Fuel burn rate = 63,500 gallons ÷ 14,430 km = 4.4 gallons per km

We need to convert km to miles, as flight time is usually measured in hours and miles. We can use the conversion factor 1 km = 0.621371 miles to convert kilometers to miles.

Range in miles = Range in km ÷ 0.621371

Range in miles = 14,430 km ÷ 0.621371 = 23,594 miles

Fuel burn rate = Fuel capacity ÷ Range in miles

Fuel burn rate = 63,500 gallons ÷ 23,594 miles = 2.69 gallons per mile

Therefore, the linear model that represents the amount of fuel on the plane as a function of flight time, in hours, is:

Fuel on plane (in gallons) = Fuel capacity - Fuel burn rate x Flight time (in hours)

F(t) = 63,500 - 2.69t

where F(t) is the amount of fuel on the plane (in gallons) after flying for t hours.

Step-by-step explanation:

Answer:

i thinks its scalene

Step-by-step explanation:

scalene is under 90 degrees

equal is 90 exact

and obtuse is over 90 but under 180

Answer: A. Action: add 3 to both sides

Step-by-step explanation:

 The property that justifies this action is the addition property of equality, which states that if you add the same number to both sides of an equation, the equation remains true.

[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3} ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\stackrel{9\times 7}{63}\\ h=6 \end{cases}\implies V=\cfrac{(63)(6)}{3}\implies V=126~cm^3[/tex]

now, for the slanted cone, we can use Cavalieri's Principle for an slanted cylinder, thus we can say that the slanted cone will have the same volume of a non-slanted cone, now, since this one has a diameter of 6, that means is has a radius of 3.

[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3} \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=3\\ h=7 \end{cases}\implies V=\cfrac{\pi (3)^2(7)}{3}\implies V=21\pi \implies V\approx 66~cm^3[/tex]

Answer:

64

Step-by-step explanation:

Answer: a = 16.62

Step-by-step explanation:

sin(37) = 10/a

a = sin(37) / 10

a = 16.62

Answer:

a ≈ 16.62

Step-by-step explanation:

using the sine ratio in the right triangle

sin37° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{10}{a}[/tex] ( multiply both sides by a )

a × sin37° = 10 ( divide both sides by sin37° )

a = [tex]\frac{10}{sin37}[/tex] ≈ 16.62 ( to 2 decimal places )

Let's use conditional probability to solve this problem. We want to find the probability of a customer purchasing a specialty espresso drink and a food item, so we can use the following formula:

P(specialty espresso drink and food) = P(food|specialty espresso drink) * P(specialty espresso drink)

We know that P(specialty espresso drink) = 0.56 and P(food|specialty espresso drink) = 0.35, so we can substitute these values into the formula:

P(specialty espresso drink and food) = 0.35 * 0.56

P(specialty espresso drink and food) = 0.196

Therefore, the probability that a randomly chosen customer will purchase a specialty espresso drink and a food item is 0.196 or approximately 0.20 (rounded to two decimal places).

So, the answer is option C: 0.20.

Answer: O: 252

Step-by-step explanation:

Theoretically, all of the letters would've been drawn 250 times as that is the theoretical possibility. The closest number to the theoretical number in this experiment would be 252 which belongs to O

Based οn the results οf the chi-square gοοdness-οf-fit test, we can cοnclude that seagulls shοw a preference fοr where they land οn the shοre.

Statistics is the branch οf mathematics dealing with data cοllectiοn, οrganizatiοn, analysis, interpretatiοn and presentatiοn.

Tο determine whether seagulls shοw a preference fοr where they land, we can cοnduct a chi-square gοοdness-οf-fit test.

First, we need tο state the null and alternative hypοtheses.

Null hypοthesis (H0): Seagulls have nο preference fοr where they land οn the shοre; the prοpοrtiοns οf seagulls landing οn sand, mud, and rοcks are equal tο the prοpοrtiοns οf sand, mud, and rοcks in the area.

Alternative hypοthesis (Ha): Seagulls have a preference fοr where they land οn the shοre; the prοpοrtiοns οf seagulls landing οn sand, mud, and rοcks are nοt equal tο the prοpοrtiοns οf sand, mud, and rοcks in the area.

We can use the fοllοwing fοrmula tο calculate the chi-square statistic:

χ² = Σ (O − E)2 / E

where χ² is the chi-square statistic, O is the οbserved frequency, E is the expected frequency, and Σ is the sum οf all cells.

The expected frequencies can be calculated as fοllοws:

We can nοw calculate the chi-square statistic:

χ² = [(128 - 112)2 / 112] + [(61 - 58)2 / 58] + [(11 - 30)2 / 30] = 8.89

Tο determine the degrees οf freedοm fοr the test, we need tο subtract 1 frοm the number οf categοries (3) tο accοunt fοr the fact that the frequencies are nοt independent. Therefοre, the degrees οf freedοm is 2.

Using a chi-square distributiοn table with 2 degrees οf freedοm and a significance level οf 0.05, we find the critical value tο be 5.99.

Since οur calculated chi-square value οf 8.89 is greater than the critical value οf 5.99, we reject the null hypοthesis. This means that seagulls dο nοt land οn the different types οf surfaces in equal prοpοrtiοns.

In cοnclusiοn, based οn the results οf the chi-square gοοdness-οf-fit test, we can cοnclude that seagulls shοw a preference fοr where they land οn the shοre.

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The value of b is -6

We know that the difference between two vectors with components (a1, a2) and (b1, b2) is given by (b1-a1, b2-a2).

So in this case, we have

[b1 - 3, b2 - 4] = [(-6) - 3, b - 4]

Simplifying the right-hand side gives us

[-9, b - 4]

So we can set the components of the left-hand side equal to the corresponding components of the right-hand side

b1 - 3 = -9

b2 - 4 = b - 4

Solving for b in the first equation gives us

b1 = -9 + 3 = -6

Substituting this into the second equation and simplifying gives us

-6 - 4 = b - 4

-10 = b - 4

b = -10 + 4 = -6

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Step-by-step explanation:

$ 540 / 12 mos  = $ 45 per month for all of the services

SO    H + D + N = 45      and    N = 2D

So     H + D + 2D   = 45

 and    H+ D = 300/12 = $25 / month

     so  (H+D) + 2D = 45

              25   + 2D = 45

                       D = 10 dollars per month

                                then H = 15 dollars per month    ( because H+D = 25)

                                        and N = 20 dollars per month  ( because N = 2D)

The indefinite integral is:

-10 ln|x| + 3 ln|x - 2| + 7 ln|x + 1| + C

First, we factor the denominator of the integrand using partial fractions:

x^3 - x^2 - 2x = (x - 2)(x + 1)x

We then express the integrand as:

10x^2 - 7x + 2 = A/(x - 2) + B/x + C/(x + 1)

Multiplying both sides by the denominator and solving for A, B, and C, we get:

A = -4, B = 10, C = 1

Thus, we can rewrite the integrand as:

-4/(x - 2) + 10/x + 1/(x + 1)

Integrating each term separately and combining the results, we obtain the final expression for the indefinite integral.

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Answer:

50/50 Chance

Step-by-step explanation:

I'm not sure if you wrote the entire question correctly, but the highest chance you would have of winning would be 100% because the 2 sides * the 50/50 chance they have. So that would make the odds of winning 50%.

Fraction of the chocolates in the box that are dark chocolate and filled with caramel is 3/32.

To solve this problem, we need to determine what fraction of chocolates in the box are both dark chocolate and filled with caramel. We are given that:

Three-fourths of the chocolates in a box are dark chocolate. Three-eighths of the dark chocolates are filled with caramel. We need to find out the fraction of chocolates that are dark chocolate and filled with caramel. To find out the fraction of chocolates in the box that are dark chocolate, we divide the number of dark chocolates by the total number of chocolates.

This is given by:

3/4 (dark chocolates)

To find the fraction of dark chocolates that are filled with caramel, we multiply the fraction of dark chocolates by the fraction of dark chocolates filled with caramel. This is given by:

3/4 × 3/8 = 9/32

Therefore, the fraction of chocolates in the box that are dark chocolate and filled with caramel is 9/32.

To simplify the fraction 9/32, we can divide the numerator and denominator by their                                             greatest common factor, which is 1:9/32 = 9 ÷ 1 / 32 ÷ 1 = 9/32.

Hence, the answer is 3/32.

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The percentage of alcohol after an hour will be approximately 4.9%.

What is percentage?

A percentage is a way to describe a part of a whole. such as the fraction 1/4 can be described as 0.25 which is equal to 25%.

To convert a fraction to a percentage, convert the fraction to decimal form and then multiply by 100 with the '%' symbol.

Let p(t) be the percentage of alcohol at time t.

Where t is in minutes.

Amount of alcohol in vat = Volume of vat x p/100

Rate of change of alcohol in vat = Volume of vat x dp/dt time 1/100

Rate of change of alcohol in vat = 500 x dp/dt x 1/100

Rate of change of alcohol in vat = 5 x dp/dt

Net rate of change of alcohol = (Rate of inflow of alcohol) - (Rate of Outflow of alcohol)

Rate of Inflow of alcohol = 0.06 - 5 = -4.94 gal/min

Rate of outflow of alcohol = p(t)/100 x 5 = 0.05p(t) gal/min

Therefore, the rate of change of alcohol is 0.03 - 0.05 p(t) gal/min

Finally we can write

5 x dp/dt = 0.3 - 0.05 p

Divide both sides by 5

dp/dt = 0.06 - 0.01p

Integrate both sides

[tex]\int {dp} / 0.06 -0.01p = \int dt[/tex]

-100ln (0.06 - 0.01p) = t + C

At t=0, p=4, therefore

-100ln (0.06 - 0.04) = 0 + C

391.2 = C

Substitute the value of C, to get

-100ln (0.06 - 0.01p) = t + 391.2

Substitute t = 60 and solve for p

-100ln (0.06 - 0.01p) = 60 + 391.2

-100ln (0.06 - 0.01p) = 451.2

ln (0.06 - 0.01p) = -4.512

[tex]p = \frac{0.06 - e^{-4.512} }{0.01}[/tex] ≈ 4.9%

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In conclusion  there are no points among the given options that are on the parabola.

why it is?

The vertex of the parabola is the midpoint between the point (6, 2) and the line y = 4, which is at (6, 3). Since the focus is below the vertex and the directrix is a horizontal line, the equation of the parabola is of the form:

(x - h)²2 = 4p(y - k)

where (h, k) is the vertex and p is the distance from the vertex to the focus (and from the vertex to the directrix).

Using the vertex and the given point (6, 2), we can find that p = 1. Therefore, the equation of the parabola is:

(x - 6)²2 = 4(y - 3)

Now we can check which of the given points satisfy this equation:

A. (4, 6)

(4 - 6)²2 = 4(6 - 3)

4 = 12 - 12

This point is not on the parabola.

B. (5, 7)

(5 - 6)²2 = 4(7 - 3)

1 = 16

This point is not on the parabola.

C. (6, 5)

(6 - 6)²2 = 4(5 - 3)

0 = 8

This point is not on the parabola.

D. (7, 6)

(7 - 6)²2 = 4(6 - 3)

1 = 12

This point is not on the parabola.

E. (8, 5)

(8 - 6)²2 = 4(5 - 3)

4 = 8

This point is not on the parabola.

Therefore, there are no points among the given options that are on the parabola.

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Complete question:

Which points are on the parabola defined as the set of points the same distance from (6,2) and the line y=4?

Answer:

ABC and CEF are right triangles | definition of right triangle

ABC and CEF are 30-60-90 triangles | definition of 30-60-90 triangle

ABC is similar to CEF | Angle-Angle-Angle similarity theorem

The measure of ∠R in ΔPQR is option (a) 26⁰

Since angle P is a right angle (90 degrees), triangle PQR is a right triangle.

Using the Pythagorean theorem, we can find the length of the remaining side PQ

PQ^2 = PR^2 - QR^2

PQ^2 = 12.7^2 - 14.1^2

PQ^2 = 161.29 - 198.81

PQ^2 = 37.52

PQ ≈ 6.12

Now we can use the sine function to find the measure of angle R

sin(R) = opposite/hypotenuse = PQ/PR

sin(R) = 6.12/12.7

sin(R) ≈ 0.482

Taking the inverse sine (sin^-1) of both sides, we get:

R ≈ sin^-1(0.482)

R ≈ 26 degrees

Therefore, the correct option is (a) 26⁰

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The given question is incomplete, the complete question is

What is the measure of angle R in triangle PQR?

a) 26°

b) 42°

c) 45°

d) 64°

Answer:

1 , 1.01 , 1.02 , 1.03 , 1.04

Step-by-step explanation:

a sequence with steps of constant size is an arithmetic sequence.

the nth term of an arithmetic sequence is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 1 and d has to be found

given a₅ = 1.04 , then

1 + 4d = 1.04 ( subtract 1 from both sides )

4d = 0.04 ( divide both sides by 4 )

d = 0.01

then sequence is

1 , 1.01 , 1.02 , 1.03 , 1.04

You need a sample size of approximately 82 students to estimate the average time college students spend studying per week with a 99% confidence interval and a margin of error of 30 minutes (0.50 hours).

To determine the sample size, we will use the formula given below;

Where, n = sample size

Zα/2 = the z-score associated with the level of confidence

α = level of significance

σ = population standard deviation

d = margin of error of the estimation.

Substituting the given values, we have; n = [(Zα/2 * σ)/d]^2

We are given the population standard deviation, σ = 3.5 hours.

We are also given the margin of error, d = 0.5 hours.

We need to find the value of Zα/2 at 99% confidence level.

Since the sample size is large enough, we can use the z-distribution to find the z-value.

The area under the standard normal distribution curve for a 99% confidence level is as shown below;

Using the z-tables, the value of Zα/2 for a 99% confidence level is 2.576.

Substituting the values into the formula; n = [(Zα/2 * σ)/d]^2n = [(2.576 * 3.5)/0.5]^2= (9.006)^2= 81.12

Since we need a whole number for the sample size, we round up the value to get 82.

However, since the standard deviation of the population is known, we can use the z-distribution to construct a confidence interval for the population mean. This will give us an accurate interval estimate of the population mean.

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The expressions x - y and x = y are sometimes a rational number.

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

x and y = rational numbers

The difference of two rational numbers is always a rational number, so x - y is sometimes a rational number.

The sum of two rational numbers is always a rational number, so x + y is sometimes a rational number.

Therefore, both expressions are sometimes a rational number.

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Answer:

False

Step-by-step explanation:

14.6/2 = 7.3

7.3 x 3 = 21.9

21.9 - 1.9 = 20

Other side:

8 - 4 = 4

4 x 3 = 12

20 does not equal 12

therefore false :)

Answer:

The answer is:

[tex]l=\dfrac{S-\pi r^2}{\pi }[/tex]

Step-by-step explanation:

In order to determine the solution for I, we have to know about the equations.

In any equation there are variables. If we want to determine the value of one of them, we have to free that variable in any side of the equation.

Regarding the surface of a cone, i have attached an image that shows the real formula of the surface of a cone, We can see that in this case:

[tex]l=rI[/tex]

So, we have the next formula and we want to l:

[tex]S=\pi l+\pi r^2[/tex]

[tex]S-\pi r^2=\pi l[/tex]

[tex]\dfrac{S-\pi r^2}{\pi }=l[/tex]

Finally the solution for I is:

[tex]l=\dfrac{S-\pi r^2}{\pi }[/tex]

The coordinates of the vertices after rotation depend on the point around which rotation is done.

Rotation of a shape around the origin (0,0) means that each point in the shape has to be moved around the origin by a certain angle. This is done by using the equation x'=xcosθ-ysinθ and y'=xsinθ+ycosθ, where x and y are the coordinates of the point before rotation and x' and y' are the coordinates of the point after rotation. In this case, the angle of rotation is 90 degrees. When we rotate triangle LMN by 90 degrees around the origin, the coordinates of the vertices become L'(2,2), M'(-4,6) and N'(4,-5).When we rotate the triangle around any of the points of L, M, or N, the coordinates of the vertices will change accordingly. For example, if we rotate the triangle around point L, the coordinates of the vertices become L'(2,-2), M'(-2,4) and N'(-6,0). Similarly, if we rotate the triangle around point M, the coordinates of the vertices become L'(2,-2), M'(6,-4) and N'(0,6). Lastly, if we rotate the triangle around point N, the coordinates of the vertices become L'(2,-2), M'(4,5) and N'(5,4).

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Step-by-step explanation:

It is a trapezoid with bases 9 and 12 ft and height ( dashed line) found using Pythagorean theorem  = 4 ft

area = height  * average of bases   =   4 * (9+12)/2 = 24 ft^2

Answer:

12x-2

Step-by-step explanation:

-4(-3x-8)-34

12x+32-34

12x-2

The experimental probability for the letter B is closest to its theoretical probability.

A number between 0 and 1 can be used tο represent prοbability, which is the likelihοοd that a given event will οccur. A prοbability οf 0 denοtes an impοssibility, while a prοbability οf 1 denοtes a certainty that the event will οccur.

By dividing the number οf balls that cοrrespοnd tο that letter by the tοtal number οf balls, οne can determine the theοretical prοbability οf drawing a letter in a single draw:

P(B) = 15/75 = 0.2

P(l) = 15/75 = 0.2

P(N) = 15/75 = 0.2

P(G) = 15/75 = 0.2

P(O) = 15/75 = 0.2

The experimental probability of drawing a letter is calculated by dividing the number of times that letter was drawn by the total number of draws:

P(B) = 247/1250 = 0.1976

P(I) = 272/1250 = 0.2176

P(N) = 238/1250 = 0.1904

P(G) = 241/1250 = 0.1928

P(Ο) = 252/1250 = 0.2016

To determine which letter's experimental probability is closest to its theoretical probability, we can calculate the difference between the two probabilities for each letter:

|P(B) - 0.2| = |0.1976 - 0.2| = 0.0024

|P(I) - 0.2| = |0.2176 - 0.2| = 0.0176

|P(N) - 0.2| = |0.1904 - 0.2| = 0.0096

|P(G) - 0.2| = |0.1928 - 0.2| = 0.0072

|P(O) - 0.2| = |0.2016 - 0.2| = 0.0016

The letter with the smallest difference is B, with a difference of 0.0024. Therefore, the experimental probability for the letter B is closest to its theoretical probability.

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Answer:

5

Step-by-step explanation:

We know that the area of the rectangle is equal to the perimeter of the triangle

S (rectangle) = (x + 8) × (2x - 1)

P (triangle) = (5x - 1) + (6x + 8) + (8x + 15)

Now we can form an equation:

(x + 8) × (2x - 1) = (5x - 1) + (6x + 8) + (8x + 15)

[tex] {2x}^{2} - x + 16x - 8 = 5x - 1 + 6x + 8 + 8x + 15[/tex]

[tex]2 {x}^{2} - 4x - 30 = 0[/tex]

[tex]d = {b}^{2} - 4 \times a \times c = ({ - 4})^{2} - 4 \times 2 \times ( - 30) = 16 + 240 = 256 > 0[/tex]

[tex]x1 = \frac{ - b - \sqrt{d} }{2 \times a} = \frac{4 - 16}{4} = \frac{ - 12}{4} = - 3[/tex]

-3 is a negative number and we cannot use it, since x must be a natural number

[tex]x2 = \frac{ - b + \sqrt{d} }{2 \times a} = \frac{4 + 16}{4} = \frac{20}{4} = 5[/tex]

The ratio commonly used as an alternative to 3.14 is 22/7 whichj commonly represents the mathematical expression of pi.

This ratio is often used as an approximation for pi (π) which is the mathematical constant representing the ratio of the circumference of a circle to its diameter. While pi is an irrational number and has an infinite number of decimal places, 22/7 is a rational number and can be easily computed.

The value of 22/7 is approximately equal to 3.1428571, which is a fairly accurate approximation for most practical applications. It is often used in situations where a quick estimate is required or where a more accurate value of pi is not necessary.

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Answer: it should be b

Step-by-step explanation: