Study the equation.
y = x2 - 4x - 77
Write the equation in graphing form (vertex form).

The total volume of the air space of spherical ball is A = 12.265625 inches³

Given data ,

Since each tennis ball has a diameter of 2.5 inches, the radius of each ball is 1.25 inches.

The air space around the balls can be thought of as a cylinder with a height equal to the diameter of one ball and a radius equal to the radius of one ball.

The height of the cylinder is 2.5 inches, and the radius is 1.25 inches.

The formula for the volume of a cylinder is:

V = πr²h

V = ( 3.14 ) ( 1.25 )² ( 7.5 )

V = 36.796875 inches³

where V is the volume, r is the radius, and h is the height.

So, the volume of the one ball is:

V₁ = ( 4/3 )π(1.25)³

V₁ = 8.177083 inches³

The total volume of three balls is = volume of 3 spherical balls

V₂ = 3V₁ = 3(8.177083) ≈ 24.53125 cubic inches

Therefore, the total volume of the air space around the three tennis balls is approximately A = 36.796875 inches³ - 24.53125 inches³

A = 12.265625 inches³

Hence , the volume of air space is A = 12.265625 inches³

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Answer:

Step-by-step explanation:

\(y=(\frac{1}{g} )x\)

Constant up a proportionally is  \(\frac{1}{g}\).

Answer:

\((x + 12)^2 + (y + 7)^2 = 277\)

Step-by-step explanation:

Equation of a circle:

The equation of a circle with center \((x_0,y_0)\) is given by:

\((x - x_0)^2 + (y - y_0)^2 = r^2\)

In which r is the radius.

Center (-12, – 7)

This means that \(x_0 = -12, y_0 = -7\). So

\((x - x_0)^2 + (y - y_0)^2 = r^2\)

\((x - (-12))^2 + (y - (-7))^2 = r^2\)

\((x + 12)^2 + (y + 7)^2 = r^2\)

Passes through the point (-3,7).

This means that we use \(x = -3, y = 7\) to find the radius squared. So

\((x + 12)^2 + (y + 7)^2 = r^2\)

\((-3 + 12)^2 + (7 + 7)^2 = r^2\)

\(81 + 196 = r^2\)

\(r^2 = 277\)

The equation of the circle is:

\((x + 12)^2 + (y + 7)^2 = r^2\)

\((x + 12)^2 + (y + 7)^2 = 277\)

Answer:

c) 42 shrubs

Step-by-step explanation:

Calculation to Find the maximum number of shrubs the garden can accommodate

First step is to calculate the Area of Garden

Area of Garden = (32.2 ft)*(18 ft)

Area of Garden= 579.6ft²

Now let calculate the maximum number of shrubs the garden can accommodate using this formula

Maximum number shrubs=(579.6 ft²)/(13.8 ft² per shrub)

Maximum number shrubs = 42 shrubs

Therefore the maximum number of shrubs the garden can accommodate will be 42 shrubs

The general solution of the partial differential equation is:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

The partial differential equation (PDE) is given as:

Uxx = (1/2)Ut with the boundary and initial conditions as 0< X <3 with U(0,t) = U(3, t)=0, U(0, t) = 5sin(4πx)

Assume that the solution can be written as a product of two functions:

U(x, t) = X(x)T(t)

Substituting this into the PDE, we have:

X''(x)T(t) = (1/2)X(x)T'(t)

Dividing both sides by X(x)T(t), we get:

(X''(x))/X(x) = (1/2)(T'(t))/T(t)

Since the left side only depends on x and the right side only depends on t, both sides must be equal to a constant, denoted as -λ²:

(X''(x))/X(x) = -λ²

(1/2)(T'(t))/T(t) = -λ²

Simplifying the second equation, we have:

T'(t)/T(t) = -2λ²

Solving the second equation, we find:

T(t) = Ce^(-2λ²t)

Applying the boundary condition U(0, t) = 0, we have:

U(0, t) = X(0)T(t) = 0

Since T(t) ≠ 0, we must have X(0) = 0.

Applying the boundary condition U(3, t) = 0, we have:

U(3, t) = X(3)T(t) = 0

Again, since T(t) ≠ 0, we must have X(3) = 0.

Therefore, we can conclude that X(x) must satisfy the following boundary value problem:

X''(x)/X(x) = -λ²

X(0) = 0

X(3) = 0

The general solution to this ordinary differential equation is given by:

X(x) = Asin(λx) + Bcos(λx)

Applying the initial condition U(x, 0) = 5*sin(4πx), we have:

U(x, 0) = X(x)T(0) = X(x)C

Comparing this with the given initial condition, we can conclude that T(0) = C = 5.

Therefore, the complete solution for U(x, t) is given by:

U(x, t) = Σ [Aₙsin(λₙx) + Bₙcos(λₙx)]*e^(-2(λₙ)²t)

where:

Σ represents the summation over all values of n

λₙ are the eigenvalues obtained from solving the boundary value problem for X(x).

To find the eigenvalues λₙ, we substitute the boundary conditions into the general solution for X(x):

X(0) = 0: Aₙsin(0) + Bₙcos(0) = 0

X(3) = 0: Aₙsin(3λₙ) + Bₙcos(3λₙ) = 0

From the first equation, we have Bₙ = 0.

From the second equation, we have Aₙ*sin(3λₙ) = 0. Since Aₙ ≠ 0, we must have sin(3λₙ) = 0.

This implies that 3λₙ = nπ, where n is an integer.

Therefore, λₙ = (nπ)/3.

Substituting the eigenvalues into the general solution, we have:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

where Aₙ are the coefficients that can be determined from the initial condition.

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Answer

a)

1939 | 103,188

1940 | 183,866

1941 | 222,202

1942 | 191,383

1943 | 191,890

1944 | 109,873

1945 | 133,792

1946 | 109,860

1947 | 156,517

1948 | 74,715

1949 | 69,479

1950 | 120, 718

1951 | 68,687

1952 | 45,030

1953 | 37,129

1954 | 60,866

1955 | 62, 786

b) 1948, 1949, 1951, 1952, 1953, 1954, 1955 all had fewer than 100,000 cases of whooping cough.

7 years out of 17 years had fewer than 100,000 cases of whooping cough.

c) Based on the data, we expect 1956 to have closer to 50,000 cases of whooping cough.

Check Explanation for more

Explanation

a) The first question asks us to arrange the table in an order that arranges them by year.

1939 | 103,188

1940 | 183,866

1941 | 222,202

1942 | 191,383

1943 | 191,890

1944 | 109,873

1945 | 133,792

1946 | 109,860

1947 | 156,517

1948 | 74,715

1949 | 69,479

1950 | 120,718

1951 | 68,687

1952 | 45,030

1953 | 37,129

1954 | 60,866

1955 | 62,786

b) Which years in this period had fewer than 100,000 cases of whooping cough?

1948, 1949, 1951, 1952, 1953, 1954, 1955 all had fewer than 100,000 cases of whooping cough.

7 years out of 17 years had fewer than 100,000 cases of whooping cough.

c) Based on the data provided, would we expect year 1956 to have closer to 50,000 cases or closer to 100,000 cases.

From the pattern of the number of whooping cough cases, year by year, we can see that the numbers decline (albeit not steadily nor consistently) year after year with only one or two years serving as serious outliers where the numbers spiked in that year, then, the decline from year to year still continues.

And seeing that none of the close years leading to 1956 had close to 100,000 cases, and in fact, this decline below 100,000 cases continues, with some years even declining below the 50,000 cases mark.

Hence, it is logical and clear to predict (bar any outrageous circumstances) that the number of expected cases of whooping cough in 1956 should be closer to the 50,000 cases mark than the 100,000 cases mark.

Hope this Helps!!!

To find the distance between Basti and Cian, we can use the law of sines in triangle ABC. The law of sines states that the ratio of the length of a side to the sine of the opposite angle is constant for all sides and their corresponding angles in a triangle.

Let's label the distance between Basti and Cian as "x". We know that the measure of angle ABC is 50 degrees and the measure of angle BAC is 60 degrees. We also know that Ali is exactly 150 ft away from Basti.

Using the law of sines, we can set up the following equation:

sin(50°) / 150 = sin(60°) / x

To solve for "x", we can rearrange the equation:

x = (150 * sin(60°)) / sin(50°)

Using a calculator, we can evaluate the expression:

x ≈ (150 * 0.866) / 0.766

x ≈ 168.4 ft

Therefore, the distance between Basti and Cian is approximately 168.4 ft.

7 x 5 + (3 x 5)      First do the parenthesis

7 x 5 + 15             Next do multiplication (left to right)

35 + 15              Last add the two remaining numbers

50

Answer : 50

This is the same as saying x = 3 and x = 7.

========================================

Work Shown:

Plug x = 3 into f(x)

f(x) = (1/2)*|x-3| + 3

f(3) = (1/2)*|3-3| + 3

f(3) = 0.5*|0| + 3

f(3) = 0.5*0 + 3

f(3) = 0 + 3

f(3) = 3

-----------------------

Plug x = 5 into f(x)

f(x) = (1/2)*|x-3| + 3

f(5) = (1/2)*|5-3| + 3

f(5) = 0.5*|2| + 3

f(5) = 0.5*2 + 3

f(5) = 1 + 3

f(5) = 4

-----------------------

Plug x = 7 into f(x)

f(x) = (1/2)*|x-3| + 3

f(7) = (1/2)*|7-3| + 3

f(7) = 0.5*|4| + 3

f(7) = 0.5*4 + 3

f(7) = 2 + 3

f(7) = 5

-----------------------

Plug x = 3 into g(x)

g(x) = sqrt(x-3) + 3

g(3) = sqrt(3-3) + 3

g(3) = sqrt(0) + 3

g(3) = 0 + 3

g(3) = 3

-----------------------

Plug x = 5 into g(x)

g(x) = sqrt(x-3) + 3

g(5) = sqrt(5-3) + 3

g(5) = sqrt(2) + 3

g(5) = 1.414214 + 3

g(5) = 4.414214 ..... which is approximate

-----------------------

Plug x = 7 into g(x)

g(x) = sqrt(x-3) + 3

g(7) = sqrt(7-3) + 3

g(7) = sqrt(4) + 3

g(7) = 2 + 3

g(7) = 5

-----------------------

To summarize everything, refer to the table below.

The rows highlighted in yellow refer to when f(x) = g(x)

This happens twice when x = 3 and x = 7

If x = 3, then f(x) = g(x) = 3

If x = 7, then f(x) = g(x) = 5

Therefore, x = 3 and x = 7 are two solutions to f(x) = g(x)

Answer:

20

Step-by-step explanation:

which side is x ? the Hypotenuse - the side opposite of the 90 degree angle ? or is it one side touching the 90 degree angle ?

that is important. we need to know which side is the Hypotenuse to make the right calculations.

if I assume x is the Hypotenuse, then the calculation follows Pythagoras like this

x² = 12² + 16² = 144 + 256 = 400

x = sqrt(400) = 20

but just in case, if x is a side touching the 90 degrees angle, then the would look like this (depending on which of the other sides is the Hypotenuse)

either

12² = 16² + x²

144 = 256 + x²

-112 = x²

that did not work. a square root of a negative number did not work for actual distances.

or

16² = 12² + x²

256 = 144 + x²

112 = x²

x = sqrt(112) = 10.58

Answer:

Let's denote:

- The number of orders Diane served as `D`

- The number of orders Sam served as `S`

- The number of orders Boris served as `B`

From the problem, we know:

1. `D + S + B = 54` (the total number of orders they served)

2. `D = S - 6` (Diane served 6 fewer orders than Sam)

3. `B = 2S` (Boris served 2 times as many orders as Sam)

We can substitute equations 2 and 3 into equation 1 to solve for the variables:

Substitute `D` and `B` in equation 1:

`(S - 6) + S + 2S = 54`

Combine like terms:

`4S - 6 = 54`

Add 6 to both sides:

`4S = 60`

Divide by 4:

`S = 15`

Now that we know `S = 15`, we can find `D` and `B` by substituting `S` into equations 2 and 3:

`D = S - 6 = 15 - 6 = 9`

`B = 2S = 2 * 15 = 30`

So, Diane served 9 orders, Sam served 15 orders, and Boris served 30 orders.

Answer:

d. -4

Step-by-step explanation:

What is the solution? 4p – 12 = 8 + 4p + 5p

a. 16

b. 6

c. -5

d. -4

Option d. -4

=> 4p – 12 = 8 + 4p + 5p

=> 4p - 4p - 5p = 8 + 12

=> -5p = 20

=> p = 20/(-5)

=> p = -4

Answer:

answer is 11: 37 am

I am sure ;-))

Answer:

\(a(1) = 144\)

\(a(n) = 144{( - \frac{1}{6} )}^{n - 1} \)

\(a(n) = - \frac{1}{6} a(n - 1)\)

Answer:

y=^3Vx is the domain function

Answer:

H₀: μ = 550 vs. Hₐ: μ < 550.

Step-by-step explanation:

A researcher believes that the mean scores for this year's college seniors in the Philippines who are interested in pursuing graduate management education will be less than 550.

The mean sore for all test‑takers is μ = 550 with a standard deviation of σ = 120.

A random sample of n = 250 college seniors in the Philippines interested in pursuing graduate management education who plan to take the GMAT were selected.

The null and alternative hypotheses for the study of the performance on the GMAT of college seniors in the Philippines is as follows:

H₀: The mean scores for this year's college seniors in the Philippines who are interested in pursuing graduate management education will not be less than 550, i.e. μ = 550.

Hₐ: The mean scores for this year's college seniors in the Philippines who are interested in pursuing graduate management education will be less than 550, i.e. μ < 550.

Answer:

0.5

Step-by-step explanation:

-20 and -40

quotient means divide one another

-20 / -40

= 20/40

= 2/4

= 0.5

I made these into tiny tiny steps, enjoy.

The translation of  the simplified algebraic expression for "the quotient of -20 and -40" is  \(\dfrac{1}{2}\).

A collection of constants, variables or numbers connected using one or more arithmetic operator is called an algebraic expression .

Example = 4y, 3x+4.

The quotient of -20 and -40 can be written as:

\(\dfrac{-20}{-40}\)

The expression can be simplified in the lowest form as;

\(\dfrac{-20}{-40}\)    =     \(\dfrac{1}{2}\)

So, the simplified algebraic expression for "the quotient of -20 and -40" is  \(\dfrac{1}{2}\).

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An exponential equation in the form \(y=a(b)^x\) that can model the amount of caffeine, y, in Kendall's body x hours after drinking the coffee is

The amount of caffeine that will be in Kendall's body after 12 hours is 18 milligrams.

In Mathematics, an exponential function can be modeled by using the following mathematical equation:

f(x) = a(b)^x

Where:

Since Kendall drank a cup of coffee that had 95 milligrams of caffeine which is decreasing at a rate of 5% per day, this ultimately implies that the relationship is geometric and the rate of change (decay rate) is given by:

Rate of change (decay rate) = 100 - 13 = 87% = 0.87.

By substituting the parameters into the exponential equation, we have the following;

\(f(x) = 95(0.87)^x\)

When x = 12, we have;

\(f(12) = 95(0.87)^{12}\)

f(12) = 17.86 ≈ 18 milligrams.

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Answer:

80

hope this is correct

comment below

hope this helps you ☺️☺️

Answer:

The correct answer would be 3) S.

Step-by-step explanation:

If you draw a line from points c to e, s is crossing paths with it. therefore, s is the answer.

Given

To find

we know that

Inserting the value of radius

Answer:

---------------------------

Given equation:

Solve it in below steps:

This equation has one solution.

Answer:

D. X=-3(y + 3) - 4

Step-by-step explanation:

The distance will be in the decimal form is :41.34

Consider the triangle :

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

According to the definition, the Pythagoras Theorem formula is given as:

\(Hypotenuse^2 = Perpendicular^2 + Base^2\)

\(c^2 = a^2 + b^2\)

We have the points are:

15,-17 and -20, -5

To find the distance between them.

The distance of x- axis is:

15 - (-20)

= 15 + 20

= 35

The distance of y- axis is:

|17 - 5| = |-22| = 22

We can now use the Pythagorean theorem (a²+b²=c²) with our imaginary triangle:

\(x^2+y^2=(distance)^2\)

\(35^2+22^2=distance^2\)

\(1709= distance^2\\Distance = \sqrt{1709}\)

In decimal form the distance would be around 41.34.

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Answer:

the data points are about evenly distributed on both sides of the line

Step-by-step explanation:

hope i helped

The simplified form of the expression is \(-32\).

To simplify the expression, we can perform the calculations written below step by step:

\(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\)\)

We follow the order of operations (PEMDAS/BODMAS):

Step 1: Simplify within parentheses:

\(\(4+6 = 10\)\).

Step 2: Perform multiplications and divisions from left to right:

\(\(-2(10) = -20\) and \(-2(4-1) = -2(3) = -6\)\).

Step 3: Evaluate the remaining additions and subtractions:

\(\(-20 + (-2) \cdot 6 = -20 - 12 = -32\)\).

Therefore, the simplified form of the expression \(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\) is \(-32\).\)

When simplifying an expression, several factors need consideration. First, apply the order of operations correctly, respecting parentheses and exponents. Next, combine like terms by adding or subtracting them. Distribute and simplify within parentheses or brackets as needed. Pay attention to negative signs and ensure their proper placement.

Finally, review the simplified expression to ensure accuracy and validity within the given context.

Note: The complete question is:

\(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\)\), calculate the simplified form of this expression.

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Answer:

y = 5

Step-by-step explanation:

\(14y-7y=35\\7y=35\\y=5\)

14 minus 7 is 7

7Y is equal to 35

divide both sides by 7 is equal to 5

Answer:

C.  

x ≈ 2.50

Step-by-step explanation:

The value of x in the given equation \(-3(-x) -6 = -3x+ 10\) is 2.66

A line graph is a type of chart used to show information that changes over time. We plot line graphs using several points connected by straight lines. We also call it a line chart. The line graph comprises of two axes known as 'x' axis and 'y' axis. The horizontal axis is known as the x-axis.

According to the given question.

We have a equation

\(-3(-x)-6 = -3x +10\)

To draw a graph for the above equation we have to simplify the given equation.

Therefore,

\(-3(-x) -6 = -3x + 10\)

⇒ \(+3x-6 = -3x + 10\)

⇒\(3x+3x = 10 + 6\)

⇒ \(6x = 16\)

⇒ \(x = \frac{16}{6}\)

⇒ \(x = 2.66\)

After solving the above equation for x we have,

x = 2.66

⇒ A line which is parallel to y axis.

Therefore, we draw a line graph at a point x = 2.66 which is parallel to y axis.

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