A parabola opening up or down has vertex (0, -4) and passes through (10, 1). Write its
equation in vertex form.

Using the derivative of the velocity, the average value of the position function from t = 2 seconds to t = 5 seconds is 14.5.

Given:

Acceleration function, a(t) = 2sin(t) + 1

Initial velocity, v(0) = 0

Initial position, s(0) = 3

To find:

Velocity function, v(t)

Position function, s(t)

Average value of the position function from t = 2 seconds to t = 5 seconds

Solution:

We know that acceleration is the derivative of velocity, and velocity is the derivative of position. So we can find the velocity and position functions by integrating the acceleration function.

Velocity function:

[tex]v(t) = \int a(t) dt\\v(t) = \int (2sin(t) + 1) dt\\v(t) = -2cos(t) + t + C1[/tex]

We know that the initial velocity, v(0) = 0. Substituting this value in the above equation, we get:

[tex]0 = -2cos(0) + 0 + C1\\C1 = 2[/tex]

Therefore, the velocity function is:

[tex]v(t) = -2cos(t) + t + 2[/tex]

Position function:

[tex]s(t) = \int v(t) dt\\s(t) = \int (-2cos(t) + t + 2) dt\\s(t) = 2sin(t) + \frac{1}{2} t^2 + 2t + C2[/tex]

We know that the initial position, s(0) = 3. Substituting this value in the above equation, we get:

[tex]3 = 2sin(0) + 0 + 0 + C2\\C2 = 3\\[/tex]

Therefore, the position function is:

[tex]s(t) = 2sin(t) + \frac{1}{2} t^2 + 2t + 3[/tex]

Average value of the position function from t = 2 seconds to t = 5 seconds:

We can find the average value of the position function using the following formula:

[tex]Avg = (1/(b-a)) * \int(a,b) f(t) dt[/tex]

Here, a = 2 and b = 5. So, substituting the values in the above formula, we get:

[tex]Avg = (1/(5-2)) * \int(2,5) (2sin(t) + \frac{1}{2} t^2 + 2t + 3) dt\\Avg = \frac{1}{3} * [ -2cos(t) + 1/6 t^3 + t^2 + 2t ] \eval(2,5)\\[/tex]

[tex]Avg = \frac{1}{3} * [ (-2cos(5) + 1/6 (5^3) + 5^2 + 25) - (-2cos(2) + 1/6 (2^3) + 2^2 + 22) ]\\Avg = 14.5[/tex]

Therefore, the average value of the position function from t = 2 seconds to t = 5 seconds is 14.5.

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The polygons are similar, therefore the missing lengths are: 12) 4; 13) 24

13. The triangles are similar by the SAS similarity theorem; 14. They are not.

Recall that every pair of corresponding sides of similar triangles are always proportional to each other.

11. Let the missing side be x.

x/24 = 6/36

x/24 = 1/6

Cross multiply

6x = 24

x = 24/6

x = 4

12. x/20 = 36/30

x/20 = 6/5

Cross multiply:

5x = 120

x = 120/5

x = 24

13. m<VWU ≅ m<SWR based on vertical angles theorem [one pair of congruent included angles]

WV/WS = 40/5 = 8

WU/WR = 40/5 = 8 [two pair of proportional sides]

Therefore, both triangles are similar by the SAS similarity theorem.

14. The triangles are not similar because:

56/24 ≠ 28/13

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Finding the total volume of a composite figure requires breaking it down into simpler shapes and then adding up their volumes.

To find the total volume of a composite figure, you'll need to break it down into simpler shapes and calculate their individual volumes.

Then, you can add up these volumes to get the total volume of the composite figure.

Here are the general steps you can follow:

Identify the simpler shapes: Look for the individual shapes that make up the composite figure, such as cubes, rectangular prisms, pyramids, cylinders, or spheres.

Calculate the volume of each shape: Use the appropriate formula for each shape to find its volume. For example, the volume of a cube is calculated by multiplying the length, width, and height of the cube.

Add up the volumes of all the simpler shapes: Once you have found the volume of each simpler shape, add them up to get the total volume of the composite figure.

Adjust for overlapping shapes: If there are any overlapping shapes, subtract the volume of the overlapped portion to avoid double-counting.

Check your work: Double-check your calculations and make sure your final answer makes sense based on the dimensions of the composite figure.

Overall, finding the total volume of a composite figure requires breaking it down into simpler shapes and then adding up their volumes.

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Approximate percent increase in attendance between the first and second game is 8.9%.

What is the approx. percentage increase in attendance between the first and second football games?

To calculate the percent increase in attendance between the first and second game, we can use the formula:

percent increase =

[(new value - old value) / old value] x 100%

where the old value is the attendance at the first game and the new value is the attendance at the second game.

Substituting the given values, we get:

percent increase = [(476 - 437) / 437] x 100%

percent increase = (39 / 437) x 100%

percent increase ≈ 8.92%

Rounding to the nearest tenth percent, we get the approximate percent increase in attendance between the first and second game as 8.9%.

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The value of the bond is multiplied by the factor [tex](1 + 0.04)[/tex], which is the same as multiplying by 1.04. After t years, the value of the bond will be multiplied by this factor t times, giving the formula:

[tex]100(1.04)t[/tex].

So, the value of the bond after t years is given by 100 • 1.04t.

A factor in mathematics is an expression or number that divides another expression or number without producing a residue. In other terms, a factor is a unit of measurement that may be multiplied by another unit of measurement to yield a specific outcome.

For example, 2 and 3 are factors of 6, because 2 multiplied by 3 equals 6. Similarly, (x + 1) and (x - 3) are factors of the expression [tex]x^2 - 2x - 3[/tex], because when these factors are multiplied together, they produce the expression:

[tex](x + 1) \times (x - 3) = x^2 - 2x - 3[/tex]

The bond is worth $100 initially, and it grows in value by 4% each year. This means that after one year, the value of the bond will be:

[tex]100 + 0.04(100) = 100(1 + 0.04) = 100(1.04)[/tex]

In other words, the value of the bond after one year is the initial value (100) multiplied by a factor of 1.04.

Similarly, after two years, the value of the bond will be:

[tex]100(1.04) + 0.04(100)(1.04) = 100(1.04)^2[/tex]

After three years, the value of the bond will be:

[tex]100(1.04)^2 + 0.04(100)(1.04)^2 = 100(1.04)^3[/tex]

In general, after t years, the value of the bond will be:

[tex]100(1.04)t[/tex]

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Following the basic principles and theory of simplifying fraction it is visible that this question follows the, the basic calculations involving the basic mathematics

Therefore the , let us take the given numerator that is 300 and then divide it using the denominator that is 25 ,

so, when we divide 300 /25 we get, the answer as12

because, when we divide both the numerator and denominator with 5 (because 25 and 300 are a divisible by 5 and are also multiples of )

we clearly see that 300 is divisible by 5 and the answer comes out to be 12.

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the composition of transformations that map ABCD to EHGF is a reflection over the y-axis followed by a translation by (4,7).

What is axis reflection ?

An axis reflection is a transformation in geometry that involves reflecting a point, a line, or an object across a given axis. In two dimensions, there are two axes of reflection: the x-axis and the y-axis.

When reflecting over the x-axis, all points stay in the same horizontal position but change their vertical position, so a point (x,y) is reflected to (x,-y).

When reflecting over the y-axis, all points stay in the same vertical position but change their horizontal position, so a point (x,y) is reflected to (-x,y).

Axis reflections can be used to transform geometric shapes such as polygons, circles, or lines. These transformations have important applications in various fields, including physics, engineering, computer graphics, and art.

To find the composition of transformations that map ABCD to EHGF, we first need to determine the individual transformations that are needed.

Reflecting over the y-axis will change the x-coordinates of the points, but leave the y-coordinates unchanged. So, to reflect over the y-axis, we negate the x-coordinates of the points.

The translation (x+4, y+7) will move each point 4 units to the right and 7 units up.

Therefore, the composition of transformations is:

Reflect over the y-axis by negating the x-coordinates:

A'(-2, 3), B'(-4, 2), C'(-4, 0), D'(-2, -1)

Translate by (4,7):

A''(2, 10), B''(0, 9), C''(0, 7), D''(2, 6)

Therefore, the composition of transformations that map ABCD to EHGF is a reflection over the y-axis followed by a translation by (4,7).

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Step-by-step explanation:

Let's assume that the monthly income of the family is x.

From the problem statement, we know that the ratio of monthly income to the monthly saving is 9:2.

Therefore, we can write:

x/4320 = 9/2

To solve for x, we can cross-multiply:

2x = 9*4320

2x = 38,880

x = 19,440

So, the monthly income of the family is Rs 19,440.

To find the monthly expenditure, we can subtract the monthly savings from the monthly income:

Monthly expenditure = Monthly income - Monthly saving

Monthly expenditure = 19,440 - 4,320

Monthly expenditure = 15,120

Therefore, the monthly expenditure of the family is Rs 15,120

Answer:

Step-by-step explanation:

[tex]2\times \frac{5}{8} = \frac{2}{1} \times\frac{5}{8}= \frac{10}{8}[/tex]

[tex]10\times \frac{1}{8} =\frac{10}{1} \times \frac{1}{8}= \frac{10}{8}[/tex]

So yes they are the same.

Answer:

To find the sum of money that amounts to Rs- 3450 in 4 months at the rate of 4.5% per annum, we can use the formula for simple interest:

Simple Interest = (Principal * Rate * Time) / 100

Where,

Principal = the sum of money borrowed or invested

Rate = the rate of interest per annum

Time = the time period in years

Since the time period given is 4 months, we need to convert it to years by dividing it by 12:

Time = 4/12 years = 1/3 years

Now, we can plug in the given values and solve for Principal:

Simple Interest = (Principal * Rate * Time) / 100

3450 = (Principal * 4.5 * 1/3) / 100

3450 * 100 = Principal * 4.5 * 1/3

11500 = Principal * 1.5

Principal = 11500 / 1.5

Principal = 7666.67 (rounded off to two decimal places)

Therefore, the sum of money that amounts to Rs- 3450 in 4 months at the rate of 4.5% per annum is Rs- 7666.67.

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Answer:

The arithmetic mean of two numbers is found by adding them together and dividing the sum by 2.

So, the arithmetic mean of 4 and 18 is:

(4 + 18) / 2 = 22 / 2 = 11

Therefore, the arithmetic mean of 4 and 18 is 11.

Step-by-step explanation:

Answer:

STEP 1:let the arithmetic mean be p, q, r hence 4,p, q, r, 18

STEP 2:U5=18=a+4d

a+4d=18

we're a=4 therefore

4+4d=18

STEP 3:4d/4=(18-4)÷4=14/4

d=14/4=3.5

p=U2=a+d=4+3.5=7.5

q=U3=a+2d=4+(2×7.5)

=

A equals (x+3) and b equals (x-3) the statement that is true for the two division problems is that a equals factor (x+3) and b equals (x-3).

For the first division problem, [tex](x^2-3x-18)[/tex]÷(x-6), the quotient is (x+3). To solve for the quotient, you first need to factor the numerator, which is (x-6)(x+3). Then, you need to divide the numerator by the denominator, which is (x-6). The quotient is (x+3). For the second division problem, [tex](x^3-x^2-5x-3)/(x^2+2x+1)[/tex], the quotient is (x-3). To solve for the quotient, you first need to factor the numerator, which is (x+1)(x-3). Then, you need to divide the numerator by the denominator, which is [tex](x^2+2x+1)[/tex]. The quotient is (x-3). Therefore, the statement that is true for the two division problems is that a equals (x+3) and b equals (x-3).

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Using the equations, the particular solution is y_p(x)=-3x+1+2e⁻ˣ.

As we already know, homogeneous equations have zero on the right side of the equation. Thus, it is said that non-homogenous differential equations are those with a function on the right side of their equation.

We are given a non-homogeneous differential equation in the form:

y′′+3y′-3y=3xe^-1​

The differential equation is called non-homogeneous because it is known to have a non-zero right-hand side.

To solve this differential equation, we first need to find the complementary function, which is the solution to the corresponding homogeneous differential equation y′′+3y′-3y=0.

We then take the first and second derivatives of y_p(x) and substitute them into the differential equation, y′′+3y′-3y=3xe^-1​ and simplify.

This leads to the system of equations:

a + 3c = 3

a + 3b - 3c = 0

a + b + 3c = 1

Solving this system of equations, we find that a=-3, b=1, and c=2. Therefore, the particular solution is y_p(x)=-3x+1+2e⁻ˣ.

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This set of equations can be solved, and the results show that a=-3, b=1, and c=2. As a result, the specific answer is [tex]y_{p(x)} =-3x+1+2e^{-x}[/tex].

Homogeneous equations, as we already know, have "0" on the right side of the equation. Therefore, it is said that differential equations with a function on the right side of the equation are non-homogenous differential equations.

A non-homogeneous differential equation of the following shape is provided to us:

[tex]y^{''} +3y^{'} -3y=3xe^{-1}[/tex]

Since the right-hand side of the differential equation is known to be non-zero, it is referred to as non-homogeneous.

Finding the complementary function, which is the answer to the related homogeneous differential equation [tex]y^{''} +3y^{'} -3y=0[/tex], is the first step in solving this differential equation.

The differential equation [tex]y^{''} +3y^{'} -3y=3xe^{-1}[/tex] is then simplified by taking the first and second derivatives of [tex]y_{p(x)}[/tex] and substituting them into

the equation.

This results in the formulae system:

a + 3c = 3

a + 3b - 3c = 0

a + b + 3c = 1

This set of equations can be solved, and the results show that a=-3, b=1, and c=2. As a result, the specific answer is  [tex]y_{p(x)} =-3x+1+2e^{-x}[/tex].

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In conclusion  the value that correctly fills in the blank in the table is 0.69.

Why it is?

To find the relative frequency of boys, we need to use the information given in the frequency table. We know that the total number of boys is 120 - 37 = 83, since the total number of students is 120 and the number of girls is 37.

We can then calculate the relative frequency for boys by dividing the number of boys who prefer math or social studies (40 + 43 = 83) by the total number of students (120):

Relative frequency for boys = (40 + 43) / 120 ≈ 0.69

Rounding to the nearest hundredth, we get:

Relative frequency for boys ≈ 0.69

Therefore, the value that correctly fills in the blank in the table is 0.69.

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Thus, from the all given expressions, the number of servings Riley will ve using is 15/2 ÷  3/4.

Once kids have a firm grasp on proper fractions, they are introduced to mixed numbers and incorrect fractions. A mixed number combines an integer (whole number) and a fraction. It is also sometimes referred to as a mixed fraction (part of a whole number).

The composite number's fractional component needs to be a legal fraction (less than one whole). The numerator (top number) of a correct fraction is less than its denominator (bottom number),

Given data:

Convert the mixed fraction 7 1/2 in the proper fraction:

= 7 1/2

= (7*2 + 1)/2

= 15/2

Number of serving cups =  Total amount of salsa / Serving cup size

Number of serving cups =  15/2 ÷  3/4

Thus, from the all given expressions, the number of servings Riley will ve using is 15/2 ÷  3/4.

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Complete question:

Riley is dividing a batch of 7 1/2 cups of salsa into equal servings that are 3/4 cup each. Select all of the expressions that can be used to find the number of servings Riley will have.

Answer: 5 [tex]\frac{7}{12}[/tex] cups

Step-by-step explanation:

   Subtract whole number values:

8 - 3 = 5 cups

   Subtract fractions:

* Get common denominators

5/6 ➜ 10/12

1/4 ➜ 3/12

* Subtract

10/12 - 3/12 = 7/12

   Answer:

5 7/12 cups

Answer:

67/12 cups of rice

Step-by-step explanation:

We Know

Jayden has 8 5/6 cups of rice.

8 5/6 = 53/6

He uses 3 1/4 cups to make dinner.

3 1/4 = 13/4

How many cups of rice does Jayden have left?

We Take

53/6 - 13/4 = 212/24 - 78/24 = 134/24 = 67/12

So, Jayden has  67/12 cups of rice left.

Answer:

The speed of the helicopter is 160mph.

Step-by-step explanation:

The speed of the helicopter is 160mph.

Speed

Speed =Distance/Time

450miles/s-35mph = 702miles/s+35

450×s+35 = 702×s-35

450s+15,750 = 702s-24,570

Collect like terms

702s-450s = 15,750 + 24,570

252s= 40,320

Divide both side by 252s

s= 40,320/252

s= 160mph

Inconclusion the speed of the helicopter is 160mph.

The value of the expression is 38 when z = 17.

Given that, an algebraic expression is described as 14 more than the product of 3 and the difference of z and 9.

First, we write the expression:

The difference of z and 9 means we subtract 9 from z. We then take this result and multiply it by 3.

Finally, we add 14 to the product of 3 and the difference of z and 9.

3(z - 9) + 14

Where z represents the value that we want to substitute into the expression.

Substituting z = 17, we get:

3(z - 9) + 14

3(17 - 9) + 14

3(8) + 14

24 + 14

38

Therefore, the value of the expression is 38.

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Answer:

Step-by-step explanation:

Answer:

y = -3/7

Step-by-step explanation:

Answer:

Step-by-step explanation:

As seen in earlier sections, the process of completing the square is a useful tool in finding noninteger values of quadratic equations, especially intercepts. When a quadratic equation of the

form f (x) = ax2

+ bx + c is put through the process of completing the square it yields an

equation of the form f (x) = a(x – h)2

+ k . The conversion of the equation to this form will

yield critical information about the equation’s characteristics before you begin to graph it.

1.) The value of h is the distance left (if negative) or right (if positive) the graph

translates from the standard position.

2.) The value of k is the distance up (if positive) or down (if negative) the graph

translates from the standard position.

3.) The values of h and k, when put together as an ordered pair, give the vertex i.e.

(h, k).

4.) The equation x = h is the formula for the axis of symmetry.

The following example demonstrates how to find the following critical information of the

equation:

a.) vertex

b.) axis of symmetry

c.) y intercept (if any)

d.) x intercepts (if any)

Example 1: Find the vertex, axis of symmetry, x-intercept(s), and y-intercept and gr

Answer:

3 hours

Step-by-step explanation:

1200/400=3

A restaurant borrows $15,700 for two months from a nearby bank. For this loan, the neighbourhood bank charges simple interest at a yearly rate of 10%. Suppose a month is one-twelfth of a year.

a) $218.06 in interest will be due after two months.

b) In the event that the restaurant doesn't pay, the balance due after two months is $15,918.06.

a) To find the interest that will be owed after 2 months, we first need to calculate the monthly interest rate:

r = (10%)/12 = 0.00833333...

We can use the formula for simple interest to find the interest owed:

I = Prt

where P is the principal (the amount borrowed), r is the interest rate per period, and t is the time in periods. Since the loan is for 2 months, we have t = 2/12 = 1/6 years.

Substituting the values, we get:

I = 15700 * 0.00833333... * (1/6) = 218.0555...

Rounding to the nearest cent, the interest owed after 2 months is $218.06.

b) The total amount owed after 2 months is the sum of the principal and the interest. Using the same values as above, we have:

The total amount owed = Principal + Interest

= 15700 + 218.0555...

= 15918.0555...

Rounding to the nearest cent, the amount owed after 2 months is $15,918.06.

The complete question is:-

A restaurant borrows $15700 from a local bank for 2 months. the local bank charges simple interest at an annual rate of 10% for this loan. assume each month is 1/12 of a year. answer each below. do not round any intermediate computations, and round your final answers to the nearest cent, if necessary, refer to the lists of financial formulas.

a) find the interest that will be owed after 2 months.

b) assuming the restaurant doesn't make any payments, find the amount owed after 2 months.

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The newspaper's claim was incorrect, as the percent error of the prediction was 0.76%, which is greater than 2%.

It is the difference between the observed value and the true value, expressed as in percentage. It is used to quantify how accurately a given measurement, prediction or other value matches the true value.

The percent error of the prediction can be calculated using the following formula:

[(Predicted Price - Actual Price) / Actual Price] × 100.

For this problem, the percent error

= [(4.10 - 4.069) / 4.069] × 100

= 0.76%.

Therefore, the newspaper's claim was incorrect since the percent error did not fall within the specified 2%.

In this case, the newspaper's prediction of the price of gasoline in two years did not match the true value as closely as expected, since the percent error was 0.76%.

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Answer:

Daisy made 50 and Oscar made 40.

Step-by-step explanation:

Daisy: Oscar: total

5            4       5+4 = 9

Taking the total made and dividing by 9.

90/9 = 10

Multiply each term in the ratio by 10.

Daisy: Oscar: total

5*10      4*10      9*10

50           40        90

Daisy made 50 and Oscar made 40.

Answer:

students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct.Trish is correct because the initial factors of 48 do not affect the prime factorization. Prime factorization is the process of breaking down a composite number into its prime factors. The prime factors of a number are those factors that are prime numbers. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, which means that 48 can be expressed as a product of these prime factors. The order in which we choose the initial factors to start the prime factorization does not affect the result, as long as we continue to break down the resulting factors into their prime factors until we cannot break them down any further. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.

Trish is correct because the prime factorization of 48 will be the same regardless of which factors are chosen as the initial factors in the tree.

To see why, let's look at the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

No matter which factors are chosen as the initial factors, the same prime factors will eventually be found.

For example, if Donna's method is used, we could start with 6 and 8:

6 = 2 × 3

8 = 2 × 2 × 2

Then we could continue to factor each of these numbers until we reach prime factors:

6 = 2 × 3

8 = 2 × 2 × 2

= 2 × 2 × 2 × 3

= 2³ × 3

Now we have found all of the prime factors of 6 and 8, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

If Larry's method is used, we could start with 4 and 12:

4 = 2 × 2

12 = 2 × 2 × 3

Then we could continue to factor each of these numbers until we reach prime factors:

4 = 2 × 2

12 = 2 × 2 × 3

= 2² × 3

Now we have found all of the prime factors of 4 and 12, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

As we can see, the same prime factors are found regardless of which factors are chosen as the initial factors in the tree. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.

The probability that on a random select today he wrote at least one check is 0.407

Let X be the no.of checks the author wrote in a day

   Average,

u = 191 checks per year

191/365                        

= 0.5233/ day

       X~Poisson ( u- 0.5233)

The p.m.f of X is given by

P(X=x) =

The probability​ that, on a randomly selected day, he wrote at least one check = P(

               = 1 - P( X < 1)

               = 1 - P(X=0)

               = 1 - e^-0.5233 x 0.5233^0/0!

               = 1 - e^-0.5233 x 1/

               = 1 - 0.59256

               = 0.407

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