one way of solving systems of linear equations is by adding a multiple of one equation to the other. the multiplier for one equation is chosen so that one of the two variables will cancel out in the sum. what should you multiply the equation y

Answer:

Max at t=2, 128 m/s

Min at t=6, 0 m/s

Step-by-step explanation:

Given the position function of a particle with respect to time, find the minimum and maximum velocity the particle travels over the interval [1,7].

[tex]s(t)=t^4-16t^3+72t^2+5[/tex]

(1) - Find the velocity function of the particle

The velocity function is a derivative of the position function.

[tex]s'(t)=v(t)\\\\s(t)=t^4-16t^3+72t^2+5\\\\\Longrightarrow s'(t)=\frac{d}{dx}[t^4-16t^3+72t^2+5] \\\\\text{Use the derivative rules.}\\\\\boxed{\left\begin{array}{ccc}\text{\underline{Power Rule:}}\\\\\frac{d}{dx}[x^n]=nx^{n-1} \end{array}\right} \ \ \boxed{\left\begin{array}{ccc}\text{\underline{Constant Rule:}}\\\\\frac{d}{dx}[k]=0 \end{array}\right} \\\\\\\Longrightarrow s'(t)=(4)t^{4-1}-16(3)t^{3-1}+72(2)t^{2-1}+0\\\\\Longrightarrow s'(t)=4t^{3}-48t^{2}+144t\\\\[/tex]

[tex]\therefore \boxed{v(t)=4t^{3}-48t^{2}+144t}[/tex]

(3) - Take the derivative of v(t)

[tex]v(t)=4t^{3}-48t^{2}+144t\\\\\Longrightarrow v'(t)=12t^2-96t+144[/tex]

(4) - Let v'(t)=0 and solve for "t," these are the critical points

[tex]v'(t)=12t^2-96t+144\\\\\Longrightarrow 0=12t^2-96t+144\\\\\Longrightarrow 0=12[t^2-8t+12]\\\\\Longrightarrow 0=t^2-8t+12\\\\\Longrightarrow (t-6)(t-2)=0\\\\\therefore \text{The critical points are} \ \boxed{t=6 \ \text{and} \ t=2}[/tex]

(5) - Find the max/min values (in this case these values represent the particle's velocity) by plugging the critical points into v(t)

[tex]\text{Recall that} \ v(t)=4t^{3}-48t^{2}+144t \ \text{and} \ t=6, \ t=2\\\\\text{\underline{When t=6:}}\\\\\Longrightarrow v(6)=4(6)^{3}-48(6)^{2}+144(6)\\\\\Longrightarrow \boxed{v(6)=0 \ m/s}\\\\\text{\underline{When t=2:}}\\\\\Longrightarrow v(2)=4(2)^{3}-48(2)^{2}+144(2)\\\\\Longrightarrow \boxed{v(6)=128 \ m/s}[/tex]

Thus, at time, t=6, the particle's velocity is smallest, 0 m/s. And at time, t=2, the particle's velocity is greatest, 128 m/s.

Answer:

f(x) = x^3 - 12x^2 + 6x - 8

f'(x) = 3x^2 - 24x + 6 = -30

3x^2 - 24x + 36 = 0

x^2 - 8x + 12 = 0

(x - 2)(x - 6) = 0, so x = 2, 6

Answer:

the values of x are :

x = 2

x = 6

The total bill of the dinner to celebrate birthday including sales tax and tip to the nearest cent is $202.01

Cost of the meal = $168.34

Sales tax = 5% of $168.34

= 0.05 × 168.34

= $8.417

Tip = 15% of $168.34

= 0.15 × 168.34

= $25.251

Total bill = Cost of the meal + Sales tax + Tip

= $168.34 + $8.417 + $25.251

= $202.008

Hence, the total bill is $202.01 to the nearest cent.

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Answer:

If $T=1.65$ seconds, then we can solve the equation for $L$ as follows:

$$

T=1.1\sqrt{L}

$$

$$

\frac{T}{1.1}=\sqrt{L}

$$

$$

\left(\frac{T}{1.1}\right)^2=L

$$

Substituting $T=1.65$, we get:

$$

L=\left(\frac{1.65}{1.1}\right)^2=2.25

$$

Therefore, the length of the pendulum is approximately 2.25 feet.

Answer:

parallel

Step-by-step explanation:

all horizontal lines are parallel

Answer:Parallel and all horizontal lines are parallel

Step-by-step explanation:

The length of the segment AB according to the given equation as required to be determined is; 32.2.

As evident from the task content; the length measure of segment AB is required to be determined.

The assumption is such that point C is the midpoint of AB.

Therefore, on this note, the triangle OBC is a right triangle and the since radius OB = 22 and OC = 15;

CB² = 22² - 15²

CB² = 484 - 225

CB² = 259

CB = 16.1

Therefore, since C is the midpoint of AB; AB = 2 × 16.1 = 32.2.

Ultimately, the length of segment AB is; 32.2.

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The value of EF is 26

A segment of a circle can be defined as a region bounded by a chord and a corresponding arc lying between the chord's endpoints.There is minor segment and major segment.

Since the two arcs are equal, i.e arc EF = arc CD, therefore,

chord EF = chord CD

9x-1 = 41 - 5x

collecting like terms

9x +5x = 41+1

14x = 42

divide both sides by 14

x = 42/14

x = 3

Therefore EF = 41-5x

= 41-5×3

= 41-15

= 26

therefore the value of EF is 26

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Let α be one root of the quadratic equation x² + sx + F = 0. If β is the other root, then we have: Therefore, the condition is that the sum of roots of the quadratic equation x² + sx + F = 0 must be zero.

Let α be one root of the quadratic equation x² + sx + F = 0. If β is the other root, then we have:

α + β = -s (sum of roots)

αβ = F (product of roots)

We are given that α is five times β, i.e., α = 5β. Substituting this in the first equation, we get:

5β + β = -s

6β = -s

β = -s/6

Using this value of β, we can find the corresponding value of α:

α = 5β = -5s/6

So, the condition for one root to be five times the other is:

α = 5β or -5s/6 = 5(-s/6)

s = 0

Therefore, the condition is that the sum of roots of the quadratic equation x² + sx + F = 0 must be zero.

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The 99% confidence interval for the population mean time spent on the internet, in minutes, is given as follows:

(48.9, 59.5).

The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

The variables of the equation are listed as follows:

The critical value, using a t-distribution calculator, for a two-tailed 99% confidence interval, with 50 - 1 = 49 df, is t = 2.68.

The parameter values are given as follows:

[tex]\overline{x} = 54.2, s = 14, n = 50[/tex]

Then the lower bound of the interval is given as follows:

[tex]54.2 - 2.68\frac{14}{\sqrt{50}} = 48.9[/tex]

The upper bound of the interval is given as follows:

[tex]54.2 + 2.68\frac{14}{\sqrt{50}} = 59.5[/tex]

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The general solution to the system x' = Ax, where A is the given matrix, can be expressed as x(t) = Ce^(At), where C is a constant matrix and e^(At) is the matrix exponential of At. This solution represents a linear combination of exponential functions, where each component of x(t) is determined by the corresponding component of C and the matrix exponential of At.

To find the general solution to the system x' = Ax, we can express the solution in terms of the matrix exponential. The matrix exponential of At, denoted as e^(At), is defined as the power series expansion of the exponential function applied to the matrix At. It can be computed using various techniques, such as diagonalization, Jordan decomposition, or power series.

The general solution to the system x' = Ax can then be written as x(t) = Ce^(At), where C is a constant matrix. This solution represents a linear combination of exponential functions, where each component of x(t) is determined by the corresponding component of C and the matrix exponential of At.

The matrix exponential e^(At) has properties that are analogous to those of the scalar exponential function. It satisfies the initial condition e^(A * 0) = I, where I is the identity matrix, and it can be used to find solutions for different initial conditions by appropriately choosing the constant matrix C.

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This means that on a random day, there is a 16.78% chance that the number of shoppers at the grocery store will fall between 200 and 400.

Using the normal distribution formula, we can calculate the z-scores for 200 and 400 shoppers:

z(200) = (200 - 505) / 115 = -2.65
z(400) = (400 - 505) / 115 = -0.91

Next, we can use a standard normal distribution table or calculator to find the area between these two z-scores. The probability is:

P(-2.65 < z < -0.91) = 0.1678

Therefore, the probability that there are between 200 and 400 shoppers at the grocery store is 0.1678.


To calculate the probability that there are between 200 and 400 shoppers at the grocery store, we first need to determine the z-scores for those values. We can then use a standard normal distribution table or calculator to find the area between those two z-scores. The result is the probability of interest. In this case, the probability that there are between 200 and 400 shoppers at the grocery store is 0.1678.


The probability that there are between 200 and 400 shoppers at the grocery store is 0.1678. This means that on a random day, there is a 16.78% chance that the number of shoppers at the grocery store will fall between 200 and 400.

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The slope of the original line of the graph sis found to be 1 and the slope of perpendicular line is -1.

One of the line shown in the graph is passing through the points (0, -7) and (7, 0). Now we have to recall that the slope M of the line L passing from points (a, b) and (c, d) is given as,

M = (d-b)/(c-a)

Also, if the line L is perpendicular to line l, then the slope m of the line l will be given as,

m = -1/M.

So, now the slope of line L is,

M = (0-(-7))/(7-0)

M = 7/7

M = 1

Now, the slope of the perpendicular line l is,

m = -1/1

m = -1.

Hence, the slope of the perpendicular line and original lines are -1 and 1.

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Answer:

17 and 35

Step-by-step explanation:

x + y = 52

x − y = 18

Add the equations together:

2x = 70

x = 35

Substitute into either equation:

y = 17

Answer:

a) 1/6. b) 1/6. c) 0. d) 1/3.

Step-by-step explanation:

it's a fair die, so probability of getting any of the 6 numbers is equal.

that is, they all have probability 1/6.

a) p(3) = 1/6

b) p(4) = 1/6

c) p(9) = 0. die only goes up to 6.

d) p(1) = 1/6. p(2) = 1/6

p(1 or 2) = 1/6 + 1/6 = 2/6 = 1/3

So the cosine of the angle between the two planes is 8sqrt(78) / 78.

To find the cosine of the angle between two planes, we need to find the normal vectors of each plane and then use the dot product formula.

The normal vector of the first plane is <1, 1, 1>, and the normal vector of the second plane is <1, 3, 4>.

The dot product of these two vectors is:

<1, 1, 1> · <1, 3, 4> = 1(1) + 1(3) + 1(4) = 8

The magnitude of the normal vector of the first plane is:

|<1, 1, 1>| = sqrt(1^2 + 1^2 + 1^2) = sqrt(3)

The magnitude of the normal vector of the second plane is:

|<1, 3, 4>| = sqrt(1^2 + 3^2 + 4^2) = sqrt(26)

Therefore, the cosine of the angle between the two planes is:

cosθ = (normal vector of plane 1) · (normal vector of plane 2) / (magnitude of normal vector of plane 1) * (magnitude of normal vector of plane 2)

= 8 / (sqrt(3) * sqrt(26))

= 8 / (sqrt(78))

= 8sqrt(78) / 78

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By solving the quadratic equations,

It will take the ball 1 second to reach 18 feet

It will take the ball 1.71 seconds to be at 10 feet

It will take the ball 2.06 seconds to hit the ground

From the question, we are to determine how long it would take the ball to reach 18 feet

To determine how long it would take the ball to reach 18 feet, we will set h = 18 in the equation

The given equation is

h = -16t² + 32t + 2

Put h = 18

18 = -16t² + 32t + 2

Rearrange

16t² - 32t - 2 + 18 = 0

16t² - 32t + 16 = 0

16t² - 16t - 16t + 16 = 0

16t(t - 1) -16(t - 1) = 0

(16t - 16)(t - 1) = 0

16t - 16 = 0 OR t - 1 = 0

16t = 16 OR t = 1

t = 16/16 OR t = 1

t = 1 OR t = 1

Hence,

t = 1 second

Hence, it will take the ball 1 second to reach 18 feet

To determine how long it will take the ball to reach 10 feet, we will set h = 10

h = -16t² + 32t + 2

Put h = 10

10 = -16t² + 32t + 2

Rearrange

16t² - 32t - 2 + 10 = 0

16t² - 32t + 8 = 0

Using the quadratic formula,

t = 0.29 second OR t = 1.71 seconds

The ball will hit the ground when h = 0

Set h = 0 in the equation

h = -16t² + 32t + 2

0 = -16t² + 32t + 2

16t² - 32t - 2 = 0

Using the quadratic formula,

t = -0.06 OR t = 2.06

Since t cannot be negative,

t = 2.06 seconds

Hence, it will take 2.06 seconds for the ball to hit the ground

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Answer:

60°

Step-by-step explanation:

ST = SW, since both are radii of the circle.

SW = WT, given.

STW is an equilateral triangle.

RED is an equilateral triangle.

Therefore, the measure of DE is 60°.

The balance after 10 years, considering monthly compounding, would be approximately $512.69.

To calculate the balance after 10 years, we can use the formula for compound interest:

A = [tex]P(1 + r/n)^{(nt)[/tex]

A is the final balance

P is the principal amount (initial deposit)

r is the annual interest rate (in decimal form)

n is the number of times the interest is compounded per year

t is the number of years

In this case, the principal amount (P) is $400, the annual interest rate (r) is 2.34% or 0.0234, the interest is compounded monthly (n = 12), and the number of years (t) is 10.

Using these values, we can calculate the final balance (A):

A = $400(1 + 0.0234/12)⁽¹²ˣ¹⁰⁾

A = $400(1.00195)¹²⁰

A ≈ $400(1.28172)

A ≈ $512.69

We may use the compound interest calculation to determine the balance after 10 years:

A =  [tex]P(1 + r/n)^{(nt)[/tex]

The ultimate balance is A.

P stands for the initial deposit's principal.

The yearly interest rate is represented by the decimal r, while the number of times it is compounded annually is represented by n.

The number of years is t.

In the above scenario, the principle (P) is $400, the annual interest rate (r) is 2.34%, or 0.0234, the interest is compounded on a monthly basis (n = 12), and the time (t) is 10.

These numbers allow us to determine the final balance (A):

A = $400(1 + 0.0234/12)⁽¹²ˣ¹⁰⁾

A = $400(1.00195)¹²⁰

A ≈ $400(1.28172)

A ≈ $512.69

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The set of numerical values that represents the minimum, lower quartile, median, upper quartile, and maximum, in that order, is:

51, 53, 55, 58, 60

To determine the minimum, lower quartile, median, upper quartile, and maximum of the given data set, we can arrange the values in ascending order:

51, 51, 53, 53, 54, 55, 55, 56, 58, 58, 58, 59, 60

Arranging the values in ascending order gives us:

51, 51, 53, 53, 54, 55, 55, 56, 58, 58, 58, 59, 60

The minimum value is 51.

The lower quartile (Q1) is the median of the lower half of the data, which is 53.

The median (Q2) is the middle value of the data set, which is 55.

The upper quartile (Q3) is the median of the upper half of the data, which is 58.

The maximum value is 60.

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There are 98,087,770 cubic feet inside the Great Pyramid.

To calculate the approximate volume of the Great Pyramid in Giza, we can use the formula for the volume of a pyramid:

Volume = (1/3) × base area × height

Since the Great Pyramid is square-based, the base area is equal to the length of one side squared. Given that the length of one side of the base is 755 feet, the base area is [tex]755^2[/tex] square feet.

Using the given height of 481 feet, we can substitute these values into the formula:

Volume = (1/3) × [tex]755^2[/tex] × 481

Calculating this expression, we find that the approximate volume of the Great Pyramid is:

Volume ≈ 98,087,770 cubic feet

There are approximately 98,087,770 cubic feet inside the Great Pyramid.

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Answer: i think

Step-by-step explanation: -9.8 m/s^2 is not the force of gravity, it is the free fall acceleration due to gravity on Earth. According to Newton's second law, F = ma. Which means that Weight = mass * gravitational

A graph representing the function g(x) = -1/2f(x + 2) is shown in the image below.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would find the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (4 - 0)/(5 - 3)

Slope (m) = 4/2

Slope (m) = 2.

At data point (3, 0) and a slope of 2, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 0 = 2(x - 3)

y = f(x) = 2x - 6

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The value of x will be 6.383cm.

Point to note

We can use the Pythagorean theorem to find the length of the diagonal which is same as diameter

diameter² = side² + side²

diameter² = 2 * x²

diameter = √(2 * x²)

Since the diameter is equal to twice the radius, we have:

diameter = 2 * radius

radius = diagonal / 2

radius = √(2 * x²) / 2

Recall that  area of the circle is given by:

area = πr²

where r is the radius

Substituting the expression for the radius, we get:

64 cm² = π * (√(2 * x²) / 2)²

64 cm² = π * (2 * x² / 4)

64 cm² = π * (x² / 2)

128 = π * x²

Solving for x, we get:

x² = 128 / π

x = √(128 / π)

x = 6.383

Therefore, the value of x is approximately 6.383cm.

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Answer:

Step-by-step explanation:

Hello!

1. Start by Plugging your numbers in. now all you have to do is 5 divided by 6! Use a calculator but anything works.

The equation for the total cost and p represent the number of pens is c = 3p + 8p + 12p

3 pens for $1.50

8 pens for $4.00

12 pens for $6.00

Where,

c represent the total cost

p represent the number of pens

Total cost, c = 1.50 + 4.00 + 6.00

= $11.50

Number of pens, p = 3 + 8 + 12

= 23

c = 3p + 8p + 12p

$11.50 = 23p

Hence, c = 3p + 8p + 12p is the equation that represents the situation.

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Answer: A C(t) = -(x-5)^2

Step-by-step explanation:

Answer:

C(t) = -x^2 + 5

Step-by-step explanation:

Notice how C(t) = 5 when t = 0. This means that (0, 5) is a point on C(t) and is the y-intercept, thus making the constant term (term without variable in standard form) 5. Next, notice how C(t) decreases as t increases and decreases. This means that C(t) is reflected over the x-axis such that it looks like an upside-down U shape. Thus, a negative must be applied to the term of the highest degree.

Answer:

Therefore, the coordinates of the midpoint M are (1, 8).

Step-by-step explanation:

The midpoint formula is:

M = ((x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints.

Using this formula, we have:

M = ((1 + 1)/2, (7 + 9)/2) = (1, 8)

Therefore, the coordinates of the midpoint M are (1, 8).

The probability of selecting a blue marble is 2/15

A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.

In this problem, to find the probability of drawing a blue marble, let's work with the sample space.

Probability of blue marble = number of blue marbles / total number of marbles

Total number of marbles = 6 + 4 + 8 + 10 + 2 = 30

Probability of blue marble = 4 / 30

probability of blue marble = 2/15 = 0.133

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Emily's constant rate in square feet per minute is 10. This means that she can paint 10 square feet in one minute.

To determine Emily's constant rate in square feet per minute, we need to use the given information that she can paint a certain number of square feet in a certain number of minutes. Let's say that Emily can paint x square feet in y minutes.
To find her rate, we need to divide the number of square feet painted by the number of minutes it took to paint them. So Emily's constant rate would be:
x / y = rate (in square feet per minute)
For example, if Emily can paint 200 square feet in 20 minutes, her rate would be:
200 / 20 = 10
Therefore, Emily's constant rate in square feet per minute is 10. This means that she can paint 10 square feet in one minute.
In conclusion, to determine Emily's constant rate in square feet per minute, we need to divide the number of square feet painted by the number of minutes it took to paint them. The result is the rate in square feet per minute.

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The cup of water reaches a temperature of 173 F after 4.5  minutes.

Temperature is a measure of the average kinetic energy of particles in a system. It is an important physical quantity used to describe the state of a system and is widely used in science, engineering, and everyday life. Temperature is a thermodynamic property of a system that shows how much energy is available to do work. In everyday terms, temperature is a measure of how hot or cold something is.

k = -0.2416

The equation for the cooling rate of the cup of water is:

T(t) = 203- 0.2416t

After 4.5 minutes, the temperature of the cup of water can be found by substituting t = 4.5 into the equation:

T(5) = 203- 0.2416(4.5 ) = 173.08 F

Therefore, the cup of water reaches a temperature of 173 F after 4.5 minutes.

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