Use Fermat's little theorem to compute the following expression.Identify the value of 3302 mod 385 using the results of previous part (a), (b), and (c) of this question and the Chinese remainder theorem.51179

i. The total height of the tent including the spire is 150 m.

ii. The length of the side of the tent  x is 132.7 m.

Trigonometric functions are required functions in determining either the unknown angle of length of the sides of a triangle.

Considering the given question, we have;

a. To determine the total height of the tent, let its height from the ground to the top of the tent be represented by x. Then:

Tan θ = opposite/ adjacent

Tan 42.7 = h/ 97.5

h = 0.9228*97.5

  = 89.97

h = 90 m

The total height of the tent including the spire = 90 + 60

                                           = 150 m

b. To determine the length of the side of the tent x, we have:

Cos θ = adjacent/ hypotenuse

Cos 42.7 = 97.5/ x

x = 97.5/ 0.7349

  = 132.67

The length of the side of the tent x is 132.7 m.

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Therefore , the solution of the given problem of standard deviation comes out to be the group standard deviation in order to use (b) z.

Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.

Here,

We use the z-distribution if the total standard deviation is known; otherwise, we use the t-distribution.

Additionally, for small sample sizes, the t-distribution is used, whereas for big sample sizes, the z-distribution is used.

The fact that z requires that you know the population standard deviation, and that this is frequently not known in practice, is one reason to use a distribution rather than the traditional .

Normal curve to find critical values when computing a level C confidence interval for a population mean.

You must be aware of the group standard deviation in order to use (b) z.

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Answer:

  (a) 0.0955

  (b) 0.0214

  (c) 0.9545

  (d) 0.3085

Step-by-step explanation:

You want the area under the standard normal PDF curve for intervals (1.22, 2.15), (2.00, 3.00), (-2.00, 2.00), and (0.50, ∞).

The probability functions of a suitable calculator or spreadsheet can find these values for you. The attachment shows one such calculator. Its "normalcdf" function takes as arguments the lower bound and upper bound.

We used 1E99 as a stand-in for "infinity" as recommended by the calculator's user manual. For the purpose here, any value greater than 10 will suffice.

Answer:

Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:

cos 2θ = 2 cos^2 θ - 1

We already know sin θ, so we can use the Pythagorean identity to find cos θ:

cos^2 θ = 1 - sin^2 θ

cos^2 θ = 1 - (2√2/5)^2

cos^2 θ = 1 - 8/25

cos^2 θ = 17/25

cos θ = ± √(17/25)

cos θ = ± (1/5) √17

Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:

cos θ = -(1/5) √17

Substituting into the double-angle identity:

cos 2θ = 2 cos^2 θ - 1

cos 2θ = 2 [-(1/5) √17]^2 - 1

cos 2θ = 2 (1/25) (17) - 1

cos 2θ = 34/25 - 1

cos 2θ = 9/25

Therefore, the exact value of cos 2θ is 9/25.

Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:

sin (θ/2) = ± √[(1 - cos θ)/2]

We already know cos θ, so we can substitute it in:

cos θ = -(1/5) √17

sin (θ/2) = ± √[(1 - cos θ)/2]

sin (θ/2) = ± √[(1 - (-1/5) √17)/2]

sin (θ/2) = ± √[(5 + √17)/10]

sin (θ/2) = ± (1/2) √(5 + √17)

Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:

sin (θ/2) = -(1/2) √(5 + √17)

Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).

The exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.

The sine of an angle in a right triangle is the ratio of the hypotenuse to the side opposite the angle.

The cosine of an angle in a right triangle is the ratio of the hypotenuse to the side adjacent the angle.

Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:

cos 2θ = 2 cos² θ - 1

Using the Pythagorean identity to find cos θ:

cos² θ = 1 - sin² θ

cos² θ = 1 - (2√2/5)²

cos² θ = 1 - 8/25

cos² θ = 17/25

cos θ = ± √(17/25)

cos θ = ± (1/5) √17

Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:

cos θ = -(1/5) √17

Substituting into the double-angle identity:

cos 2θ = 2 cos² θ - 1

cos 2θ = 2 [-(1/5) √17]² - 1

cos 2θ = 2 (1/25) (17) - 1

cos 2θ = 34/25 - 1

cos 2θ = 9/25

Therefore, the exact value of cos 2θ is 9/25.

Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:

sin (θ/2) = ± √[(1 - cos θ)/2]

We already know cos θ, so we can substitute it in:

cos θ = -(1/5) √17

sin (θ/2) = ± √[(1 - cos θ)/2]

sin (θ/2) = ± √[(1 - (-1/5) √17)/2]

sin (θ/2) = ± √[(5 + √17)/10]

sin (θ/2) = ± (1/2) √(5 + √17)

Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:

sin (θ/2) = -(1/2) √(5 + √17)

Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).

Hence, the exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.

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Answer:

the question is incomplete, so I looked for similar questions:

There are 3 sandwiches, 4 drinks, and 2 desserts to choose from.

the answer = 3 x 4 x 2 = 24 possible combinations

Explanation:

for every sandwich that we choose, we have 4 options of drinks and 2 options of desserts = 1 x 4 x 2 = 8 different options per type of sandwich

since there are 3 types of sandwiches, the total options for lunch specials = 8 x 3 = 24

If the numbers are different, all we need to do is multiply them. E.g. if instead of 3 sandwiches there were 5 and 3 desserts instead of 2, the total combinations = 5 x 4 x 3 = 60.

For this question's answer, there are 2 x 5 = 10 lunch specials are possible.

The number of lunch specials possible are 10.

We can use combinations for this case,

Total number of distinguishable things is m.

Out of those m things, k things are to be chosen such that their order doesn't matter.

This can be done in total of

[tex]^mC_k = \dfrac{m!}{k! \times (m-k)!} ways.[/tex]

If the order matters, then each of those choice of k distinct items would be permuted k! times.

So, total number of choices in that case would be:

[tex]^mP_k = k! \times ^mC_k = k! \times \dfrac{m!}{k! \times (m-k)!} = \dfrac{m!}{ (m-k)!}\\\\^mP_k = \dfrac{m!}{ (m-k)!}[/tex]

This is called permutation of k items chosen out of m items (all distinct).

We are given that;

Number of sandwiches=2

Number of drinks=5

Now,

To find the total number of lunch specials, we need to multiply the number of choices for sandwiches by the number of choices for drinks.

Number of sandwich choices = 2

Number of drink choices = 5

Total number of lunch specials = 2 x 5 = 10

Therefore, by combinations and permutations there are 10 possible lunch specials.

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Answer: False

Step-by-step explanation:

25% of the data is between Q1 and the median.

According to one meaning of the phrase, it merely refers to the selling price of something. For instance, a piece of art would be sold for that amount if bids reached a record high of $10 million. Thus, option A is correct

The sale price of bananas this week is 10% off the original price of $1.40 per pound, which means the sale price is:

$1.40 - 10% of $1.40 = $1.26 per pound

To find the total cost of 6 pounds of bananas this week, we can multiply the sale price by the number of pounds:

$1.26 per pound * 6 pounds = $ [tex]7.56[/tex]

Therefore, the total price of 6 pounds of bananas this week at the grocery store is $ [tex]7.56[/tex] .

However, we need to be careful with the answer choices provided. They all differ from $7.56, so we need to double-check our calculations.

If we add a 10% discount to $1.40 per pound, we get:

$1.40 - (10/100)*$1.40 = $1.26 per pound

And the total cost of 6 pounds at $1.26 per pound is:

$ [tex]1.26 \times 6[/tex] = $ [tex]7.56[/tex]

Therefore,  $ [tex]8.19[/tex] is not a possible answer, and the other options are either miscalculated or rounded incorrectly.

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Answer: 5 standard glasses

A standard glass of wine is 5 ounces. This is typical for any dry white, red, orange, or rosé wine. A standard bottle is 750mL, or about 25 ounces of wine. So, a normal 750mL bottle has 5 standard glasses of wine.

200 people invested in neither stocks nor bonds for retirement.

The inclusion-exclusion principle is a counting method used to determine the size of a set created by joining two or more sets. It is predicated on the notion that if we just sum the set sizes, we can wind up counting certain components more than once (the elements that are in the intersection of the sets). We deduct the sizes of the sets' intersections from the sum of their sizes to prevent double counting.

The total number of people who invested in stocks are:

Total = Stocks + Bonds - Both

Total = 700 + 400 - 300

Total = 800

Using the inclusion- exclusion principle:

neither = Total surveyed - Total

neither = 1000 - 800 = 200

Hence, 200 people invested in neither stocks nor bonds for retirement.

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Cardinality of given set is 10.

The cardinality of a mathematical set refers to the number of entries in the set. It may be limited or limitless. For instance, if set A has six items, its cardinality is equivalent to 6: 1, 2, 3, 4, 5, and 6. A set's size is often referred to as the set's cardinality. The modulus sign is used to indicate it on either side of the set name, |A|.

A set that can be counted and has a finite number of items is said to be finite. On the other hand, an infinite set is one that has an unlimited number of components and can either be countable or uncountable.

Possible set of A=14+4+1+9=28

Possible set of C=1 +6+9+9=25

n(A∩ C)=10

Hence, Cardinality of given set is 10.

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Therefore, the equation of the tangent line at (5,f(5)) is y = 18x - 65.

In mathematics, the slope of a line is a measure of its steepness or incline, usually denoted by the letter m. It describes the rate of change of a line in the vertical direction compared to the horizontal direction. The slope of a line can be positive, negative, zero, or undefined, depending on the angle it makes with the horizontal axis. The slope of a line is commonly calculated as the ratio of the change in the y-coordinates to the change in the x-coordinates between any two points on the line.

Here,

(A) The slope of the secant line joining (2,f(2)) and (7,f(7)) is given by:

slope = (f(7) - f(2)) / (7 - 2)

We can find f(7) and f(2) by substituting 7 and 2, respectively, into the function f(x):

f(7) = 7² + 8(7) = 49 + 56 = 105

f(2) = 2² + 8(2) = 4 + 16 = 20

Substituting these values into the formula for the slope of the secant line, we get:

slope = (105 - 20) / (7 - 2) = 85 / 5 = 17

Therefore, the slope of the secant line joining (2,f(2)) and (7,f(7)) is 17.

(B) The slope of the secant line joining (5,f(5)) and (5+h,f(5+h)) is given by:

slope = (f(5+h) - f(5)) / (5+h - 5) = (f(5+h) - f(5)) / h

We can find f(5) and f(5+h) by substituting 5 and 5+h, respectively, into the function f(x):

f(5) = 5² + 8(5) = 25 + 40 = 65

f(5+h) = (5+h)² + 8(5+h) = 25 + 10h + h² + 40 + 8h = h² + 18h + 65

Substituting these values into the formula for the slope of the secant line, we get:

slope = ((h² + 18h + 65) - 65) / h = h² / h + 18h / h = h + 18

Therefore, the slope of the secant line joining (5,f(5)) and (5+h,f(5+h)) is h+18.

(C) The slope of the tangent line at (5,f(5)) is equal to the derivative of the function f(x) at x=5. We can find the derivative of f(x) as follows:

f(x) = x² + 8x

f'(x) = 2x + 8

Substituting x=5, we get:

f'(5) = 2(5) + 8 = 18

Therefore, the slope of the tangent line at (5,f(5)) is 18.

(D) The equation of the tangent line at (5,f(5)) can be written in point-slope form as:

y - f(5) = m(x - 5)

where m is the slope of the tangent line, which we found to be 18. Substituting the values of m and f(5), we get:

y - 65 = 18(x - 5)

Simplifying, we get:

y = 18x - 65

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There are 792 strings across a,b, and 27,720 in a,b,c.

(a) We must select five slots for a's in an alphabet of "a,b" before filling the remaining spaces with "b's." Hence, the binomial coefficient is what determines how many strings of length 12 that include precisely five as:

C(12,5) = 792

As a result, there are 792 distinct strings of length 12 that include exactly five a's across the letters a, b.

(b) We may use the same method as before for an alphabet consisting of the letters "a,b,c." The first five slots must be filled with a's, followed by three b's, and the final four positions must be filled with c's. The number of strings of length 12 that contain exactly five a's across the letters "a," "b," and "c" is thus given by:

C(12,5) * C(7,3) = 792 * 35 = 27720

Thus, there are 27,720 distinct strings.

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In linear equation, 11.85 pounds is the weight of the wire.

What is  linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

Total weight of pool having 16 wires     =13.6 pounds

Weight of the pool                                 =1.75

Therefore the weight of the wire alone = 13.6 - 1.75

                                                             =  11.85 pounds

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a) F(x,y,z) = y'z'. b) F(x,y,z) = x'y'z' + x'y'z + xy'z'. c) F(x,y,z) = x'y'z'. d) F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z'. These are the sum-of-products expansions of the Boolean function F(x, y, z) for the given conditions.

a) When x = 0, F(x,y,z) equals 1 if and only if yz = 0. This can be expressed as the sum of products: F(x,y,z) = y'z' (read as "not y and not z").

b) When xy = 0, F(x,y,z) equals 1 if and only if either x = 0 or y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' + x'y'z + xy'z' (read as "not x and not y and not z" OR "not x and not y and z" OR "x and not y and not z").

c) When x + y = 0, F(x,y,z) equals 1 if and only if x = y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' (read as "not x and not y and not z").

d) When xyz = 0, F(x,y,z) equals 1 if and only if x = 0 or y = 0 or z = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z' (read as "not x and not y and z" OR "not x and y and not z" OR "x and not y and not z" OR "not x and not y and not z").

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Answer:

Traffic signs are regulated by the Manual on Uniform Traffic Control Devices (MUTCD). The perimeter of a rectangular traffic sign is 126 inches. Also, its length is 9 inches longer than its widthFind the dimensions of this sign.

Step-by-step explanation:

Let's say the width of the sign is x inches. Then, according to the problem, the length of the sign is 9 inches longer than the width, which means the length is x + 9 inches.

The perimeter of a rectangle can be found by adding up the length of all its sides. For this sign, the perimeter is given as 126 inches. So we can set up an equation:

2(length + width) = 126

Substituting the expressions for length and width in terms of x, we get:

2(x + x + 9) = 126

Simplifying and solving for x:

2(2x + 9) = 126

4x + 18 = 126

4x = 108

x = 27

So the width of the sign is 27 inches, and the length is 9 inches longer, or 36 inches. Therefore, the dimensions of the sign are 27 inches by 36 inches.

The moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 elsewhere is M(t) = 1, and its first and second central moments, μ1′ and μ2′, and the variance, σ2, are 0, 0 and 0 respectively.

The moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 elsewhere is M(t) = 1.

Using M(t) we can calculate the first and second central moments, μ1′ and μ2′, and the variance, σ2, as follows:

μ1′ = M′(t) = 0

μ2′ = M′′(t) = 0

σ2 = μ2′ - (μ1′)2 = 0 - (0)2 = 0.

Therefore, the first and second central moments, μ1′ and μ2′, and the variance, σ2, of the continuous random variable X with probability density f(x) = 1 for 0 elsewhere are 0, 0 and 0 respectively.

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Answer:

See below.

Step-by-step explanation:

We can solve this linear programming problem using the simplex method. We will start by converting the problem into standard form

Maximize z = 3x₁ + 5x₂ + 0s₁ + 0s₂

subject to

x₁ - 5x₂ + s₁ = 35

3x₁ - 4x₂ + s₂ = 21

x₁, x₂, s₁, s₂ ≥ 0

Next, we create the initial tableau

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

We can see that the initial basic variables are s₁ and s₂. We will use the simplex method to find the optimal solution.

Step 1: Choose the most negative coefficient in the bottom row as the pivot element. In this case, it is -5 in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

Step 2: Find the row in which the pivot element creates a positive quotient when each element in that row is divided by the pivot element. In this case, we need to find the minimum positive quotient of (35/5) and (21/4). The minimum is (21/4), so we use the second row as the pivot row.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 4/5 0 1/5 1 28/5

x₂ -3/4 1 0 -1/4 -21/4

z 39/4 0 15/4 3/4 105

Step 3: Use row operations to create zeros in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 0 1/4 7/20 49/10

x₂ 0 1 3/16 -1/16 -21/16

z 0 0 39/4 21/4 525/4

The optimal solution is x₁ = 49/10, x₂ = 21/16, and z = 525/4.

Therefore, the maximum value of z is 525/4, which occurs when x₁ = 49/10 and x₂ = 21/16.

Answer:

5 as a percentage of 100 is 5/100 which is 5%

Answer: 2/13

Step-by-step explanation:

There are four jacks and four tens in a standard deck of 52 cards. However, the jack of spades and the ten of spades are counted twice since they are both a jack and a ten. Therefore, there are 8 cards that are either a jack or a ten, and the probability of drawing one of these cards at random is:

P(Jack or Ten) = 8/52 = 2/13

So the answer is 2/13.

Step-by-step explanation:

a probability is airways the ratio

desired cases / totally possible cases

in each of the 4 suits there is one Jack and one 10.

that means in the whole deck of cards we have

4×2 = 8 desired cases.

the totally possible cases are the whole deck = 52.

so, the probability to draw a Jack or a Ten is

8/52 = 2/13

Answer:

Fraction: 241/48

Improper fraction: 5 1/48

Decimal: 5.021

Step-by-step explanation:

To calculate the expression 13/16+2 1/12+2 3/24, I first converted the mixed numbers 2 1/12 and 2 3/24 to improper fractions. 2 1/12 is equal to 25/12 and 2 3/24 is equal to 51/24. Then, I added the three fractions 13/16, 25/12, and 51/24 by finding a common denominator, which in this case is 48. So, the expression becomes (39/48)+(100/48)+(102/48), which simplifies to (39+100+102)/48, which equals 241/48. Finally, I converted the improper fraction 241/48 to a mixed number, which is equal to 5 1/48.

To obtain the required equation we divide the equation by sin²Ø.

The first six functions are trigonometric, with the domain value being the angle of a right triangle and the range being a number. The angle, expressed in degrees or radians, serves as the domain and the range of the trigonometric function (sometimes known as the "trig function") of f(x) = sin. Like with all other functions, we have the domain and range. In calculus, geometry, and algebra, trigonometric functions are often utilised.

The given equation is:

sin²Ø+cos²Ø=1

To obtain the required equation we divide the equation with sin²Ø:

sin²Ø/sin²Ø +cos²Ø/ sin²Ø = 1/sin²Ø

1 + cot²Ø = csc²Ø

Hence, to obtain the required equation we divide the equation by sin²Ø.

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Answer:

Required length is 13 feet

Step-by-step explanation:

[tex]{ \rm{length = \sqrt{ {12}^{2} + {5}^{2} } }} \\ \\ { \rm{length = \sqrt{144 + 25} }} \\ \\ { \rm{length = \sqrt{169} }} \\ \\ { \rm{length = 13 \: feet}}[/tex]

Answer:

Step-by-step explanation:

long leg = 78  (means that 26√3*√3 = 26√9 = 26*3 = 78

for x: Short leg= 26√3

Hypotenuse= 2*26√3 = 52√3 for y

Answer:

3. One solution

Step-by-step explanation:

-x²-9 = 6x

or, x²+6x+9 = 0

or, x²+2.x.3+3² = 0 [using (a+b)² = a²+2ab+b²]

or, (x+3)² = 0

or, x+3 = 0

x = -3

The equation of given line which is graphed is  [tex]x+2=0.[/tex] By locating the slope (m) and y-intercept (b) in the graph of a line, we can define a linear function in the form y=mx+b.

A straight line's graph equation can be expressed as [tex]y = m x + c[/tex]  , which consists of a term, a term, and a number. a new. to the a and the likes in the likes thes of the likes of thes of thes of thes of thes of thes of people.

A line graph is a type of graph that uses straight lines to connect the data points. A line graph can show how something changes over time or compares different situations1.

A horizontal line has the equation \(y = c\), where \(c\) is a constant. This means that the \(y\)-value of every point on the line is the same

Therefore, The set of all points (x,y) in the plane that satisfy the equation   [tex]y=f(x) y = f (x)[/tex]    is the function's graph.

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The correct standard form of the equation of the parabola is:

[tex]y = -x^2 - 1[/tex].

To find the standard form of the equation of the parabola that passes through the given points (-3, 3) and (1, -1), we can use the general form of the equation of a parabola:

[tex]y = ax^2 + bx + c[/tex] ___________(1)

Substituting the coordinates of the two given points into this equation, we get a system of two equations in three unknowns (a, b, and c):

[tex]3 = 9a - 3b + c[/tex]

[tex]-1 = a + b + c[/tex]

To solve for a, b, and c, we can eliminate one of the variables using subtraction or addition. Subtracting the second equation from the first, we get:

[tex]4 = 8a - 4b[/tex]

Simplifying this equation, we get:

[tex]2 = 4a - 2b[/tex]

Dividing both sides by 2, we get:

[tex]1 = 2a - b[/tex]___________(2)

Now we can substitute this expression for b into one of the earlier equations to eliminate b. Using the first equation, we get:

[tex]3 = 9a - 3(2a - 1) + c[/tex]

Simplifying this equation, we get:

[tex]3 = 6a + c + 3[/tex]

Subtracting 3 from both sides, we get:

[tex]0 = 6a + c[/tex]

Solving for c, we get:

c = -6a __________(3)

Substituting this expression for c into the second equation, we get:

[tex]-1 = a + (2a - 1) - 6a[/tex]

Simplifying this equation, we get:

[tex]-1 = -3a - 1[/tex]

Adding 1 to both sides, we get:

[tex]-3a =0[/tex]

Solving for a, we get:

[tex]a = 0[/tex]

Substituting this value of a into the equation(3) for c, we get:

c = 0

Substituting a = 0 into the equation(2) for b that we found earlier, we get:

[tex]1 = 0 - b[/tex]

Solving for b, we get:

[tex]b = -1[/tex]

Putting the values of a, b and c in (1), we get

[tex]y = -x^2 - 1[/tex]

Therefore, the equation of the parabola that passes through the given points (-3, 3) and (1, -1) is:

[tex]y = -x^2 - 1[/tex]

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Complete question:

A parabola goes through (-3, 3) & (1, -1). A point is below the parabola at (-3, 2). A line above the parabola goes through (-3, 4) & (0, 4). A point on the parabola is labeled (x, y).

What is the correct standard form of the equation of the parabola?

The figure is in the image attached below

Answer:

The value of the derivative at (0, 0) is zero.

Step-by-step explanation:

Given function:

[tex]f(x)=\dfrac{x^2}{x^2+4}[/tex]

To differentiate the given function, use the quotient rule and the power rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Quotient Rule of Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Let}\;u &= x^2& \implies \dfrac{\text{d}u}{\text{d}{x}} &=2 \cdot x^{(2-1)}=2x\\\\\textsf{Let}\;v &=x^2+4& \implies \dfrac{\text{d}v}{\text{d}{x}} &=2 \cdot x^{(2-1)}+0=2x\end{aligned}[/tex]

Apply the quotient rule:

[tex]\implies f'(x)=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}[/tex]

[tex]\implies f'(x)=\dfrac{(x^2+4) \cdot 2x-x^2 \cdot 2x}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{2x(x^2+4)-2x^3}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{2x^3+8x-2x^3}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{8x}{(x^2+4)^2}[/tex]

An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (0, 0).

To determine the value of the derivative at the minimum point, substitute x = 0 into the differentiated function.

[tex]\begin{aligned}\implies f'(0)&=\dfrac{8(0)}{((0)^2+4)^2}\\\\&=\dfrac{0}{(0+4)^2}\\\\&=\dfrac{0}{(4)^2}\\\\&=\dfrac{0}{16}\\\\&=0 \end{aligned}[/tex]

Therefore, the value of the derivative at (0, 0) is zero.

The the inverse of [tex]n = \frac{3t+8}{5}$[/tex] is [tex]t = \frac{5n-8}{3}$[/tex].

To find the inverse of [tex]n = \frac{3t+8}{5}$[/tex], we need to solve for t in terms of n.

Starting with the given equation, we can first multiply both sides by 5 to get rid of the fraction:

[tex]$$5n = 3t + 8$$[/tex]

Next, we can isolate t by subtracting 8 from both sides and then dividing by 3:

[tex]$\begin{align*}5n - 8 &= 3t \\frac{5n-8}{3} &= t\end{align*}[/tex]

Therefore, the inverse of n is:

[tex]$t = \frac{5n-8}{3}$$[/tex]

We can also check that this is indeed the inverse by verifying that:

[tex]$n = \frac{3t+8}{5} = \frac{3}{5} \cdot \frac{5n-8}{3} + \frac{8}{5} = n$$[/tex]

So, the inverse of [tex]n = \frac{3t+8}{5}$[/tex] is [tex]t = \frac{5n-8}{3}$[/tex].

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The Butler family used their sprinkler for 30 hours and the Phillips family used their sprinkler for 25 hours.

Let's solve the problem with algebra.

Let x represent the number of hours the Butlers used their sprinkler, and y represent the number of hours the Phillips family used their sprinkler. We are aware of the following:

The Butler family's sprinkler had a water output rate of 25 L per hour, so the total amount of water they used is 25x.

The Phillips family's sprinkler had a water output rate of 40 L per hour, so the total amount of water they used was 40y.

The sprinklers were used by the families for a total of 55 hours, so x + y = 55.

The total amount of water produced was 1750 L, so 25x + 40y = 1750.

Using these equations, we can now solve for x and y.

First, we can solve for one of the variables in terms of the other using the equation x + y = 55. For instance, we can solve for x as follows:

x = 55 - y

When we plug this into the second equation, we get:

25(55 - y) + 40y = 1750

We get the following results when we expand and simplify:

1375 - 25y + 40y = 1750

15y = 375

y = 25

As a result, the Phillips family ran their sprinkler for 25 hours. We get the following when we plug this into the equation x + y = 55:

x + 25 = 55

x = 30

As a result, the Butlers used their sprinkler for 30 hours.

As a result, the Butler family sprinkled for 30 hours and the Phillips family sprinkled for 25 hours.

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Distance from a point in space to the moon is a continuous random variable and Number of coins in a stack is a discrete random variable.

A discrete random variable is one that has a finite number of possible values or one that can be countably infinitely numerous. A discrete random variable is, for instance, the result of rolling a die because there are only six possible outcomes.

A continuous random variable, on the other hand, is one that is not discrete and "may take on uncountably infinitely many values," like a spectrum of real numbers.

1. The Name of the people in the car that crosses the bridge - Not a Variable

2. Continuous random variable measuring the interval between each car crossing the bridge.

3. The Categorical Random Variable for the type of vehicles crossing the bridge

4. The number of cars that use the bridge in one hour - Continuous Random Variable

For Question 2:

1. Type of coin - Categorical Random Variable

2. The distance from a given location in space to the moon - Continuous Random Variable

3. Number of coins in a stack - Discrete Random Variable

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Complete question is:

Label each of the following as Discrete Random Variable, Continuous Random Variable, Categorical Random Variable, or Not a Variable.

1. The Name of the people in the car that crosses the bridge Not a Variable

2. The time between each car crossing the bridge Continuous Random Variable

3. The type of cars that cross the bridge Categorical Random Variable

4. The number of cars that use the bridge in one hour Continuous Random Variable

Question 2: Which of these are Continuous and which are Discrete Random Variables?

1. Type of coin

2. Distance from a point in space to the moon

3. Number of coins in a stack