A model rocket is launched with an initial upward velocity of 105 ft/s. The rocket’s height “h” (in feet) after “t” seconds is given by the following. h=105t-16t^2. Find all values of t for which the rocket’s height is 34 feet.

Based on the formula of the quadratic function y=-x^2+6x-5, we know that its graph is a downward-facing parabola that opens wide, with a vertex at (3,-14), and an axis of symmetry at x=3.

Based on the formula of the quadratic function y=-x^2+6x-5, we can determine several properties of its graph, including its shape, vertex, and axis of symmetry.

First, the negative coefficient of the x-squared term (-1) tells us that the graph will be a downward-facing parabola. The leading coefficient also tells us whether the parabola is narrow or wide. Since the coefficient is -1, the parabola will be wide.

Next, we can find the vertex using the formula:

Vertex = (-b/2a, f(-b/2a))

where a is the coefficient of the x-squared term, b is the coefficient of the x term, and f(x) is the quadratic function. Plugging in the values for our function, we get:

Vertex = (-b/2a, f(-b/2a))

= (-6/(2*-1), f(6/(2*-1)))

= (3, -14)

So the vertex of the parabola is at the point (3,-14).

Finally, we know that the axis of symmetry is a vertical line passing through the vertex. In this case, it is the line x=3.

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The general solution to the given initial value problem is z(x) = A sin(x) + B cos(x) + \(e^{-6x}\)/6, where A and B are constants determined by the initial conditions.

The given equation z''(x) + z(x) = 4\(e^{-6x}\) is a nonhomogeneous linear second-order differential equation. To find the general solution, we first solve the associated homogeneous equation, which is z''(x) + z(x) = 0. The characteristic equation is\(r^2\) + 1 = 0, which has complex roots r = ±i. Therefore, the homogeneous solution is z_h(x) = A sin(x) + B cos(x), where A and B are arbitrary constants.

To find a particular solution to the nonhomogeneous equation, we assume a solution of the form z_p(x) = C\(e^{-6x}\), where C is a constant to be determined. Plugging this into the differential equation, we have C(-36\(e^{-6x}\)) + C\(e^{-6x}\) = 4\(e^{-6x}\). Equating the coefficients of \(e^{-6x}\), we find C = 4/(-36+1) = -4/35.

The general solution is then given by the sum of the homogeneous and particular solutions: z(x) = z_h(x) + z_p(x) = A sin(x) + B cos(x) - (4/35)\(e^{-6x}\).

Applying the initial conditions z(0) = 0 and z'(0) = 0, we have A + B - 4/35 = 0 and A - 4/35 = 0. Solving these equations, we find A = 4/35 and B = 4/35.

Therefore, the final solution to the initial value problem is z(x) = (4/35)sin(x) + (4/35)cos(x) - (4/35)\(e^{-6x}\).

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The fraction of 2 hrs 15 minutes to 1/4 of an hour is 1/9.

What is fraction?

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

Since,

1 hour = 60 minutes.

2 hours = 120 minutes.

2 hours 15 minutes = 120 + 15 = 135 minutes.

1/4 of an hour = 60/4 = 15 minutes.

So, the fraction is,

\(\frac{15}{135} = \frac{1}{9}\)

Therefore, the fraction of 2 hrs 15 minutes to 1/4 of an hour is 1/9.

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Answer:

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

Step-by-step explanation:

Answer:

-9=n

Step-by-step explanation:

-13=4-n

+4 to both sides to get n by its self

-13+4=9

-13-4=9

-9=n

Answer:

64 ft^2

Step-by-step explanation:

It is a square to start, so each side = x

then     (x+7 )    *    ( x -2)    = 90

          x^2 + 5x -14 = 90

           x^2 + 5x - 104 = 0         factor or use Quadratic Formula to fin x = 8 ft

Original area is then 8 x 8 = 64 ft^2  

Answer:

B I'm thinking, I have to answer any question so I might be wrong here, also eat some salad it's nasty I know

Answer:

option c. ±\( \sqrt{65} \)

Step-by-step explanation:

In the given right angled triangle we can see that :-

by Pythagoras theorem

hypotenuse² = adjacent ² + opposite ²

9² = 4² + x²

81 = 16 + x²

combining like terms

81 - 16 = x²

65 = x²

x = ±\( \sqrt{65} \)

5. The constant of proportionality is 1.5

The equation is p = 1.5×s

6. The constant of proportionality is 1.99

The equation is p = 1.99 × a

7. The variables Number of Pens and Cost  are not proportional

Please find attached the required graph

8. The variables Number of minutes and Words Typed are not proportional

Please find attached the required graph

The procedure for finding the answers are as follows;

5. The given data are presented as follows;

\(\begin{array}{ccc}Shirts \ (s)&&Profit \ (p)\\5&&7.50\\10&&15.00\\15&&22.50\end{array}\)

Where two variables, s and p are proportional, we get;

p ∝ s

Therefore;

p = C × s

C = p/s

Where;

C = The constant of proportionality

Therefore, the constant of proportionality, C, of the given variables, (number of shirts, s, and profit, p, is found as follows;

C = 7.50/5 = 15.00/10 = 22.50/15 = 1.5

The constant of proportionality, C = 1.5

The equation that relates the two values is p = 1.5×s

6.  For the apples to price relationship, we have;

\(\begin{array}{ccc}Apples \ (a)&&Price\ (p)\\4&&7.96\\5&&9.95\\6&&11.94\end{array}\)

Therefore;

p ∝ a

p = C × a

C = p/a

Plugging in the values gives;

C = 7.96/4 = 9.95/5 = 11.94/6 = 1.99

The constant of proportionality, C = 1.99

Therefore, the equation relating the two values is p = 1.99 × a

7. The given data is presented in a tabular form as follows;

\(\begin{array}{ccc}Number \ of \ pens &&Cost\ \\1&&52\\2&&54\\3&&56\\4&&58\end{array}\)

A set of data is proportional or has a proportional relationship if their x, and therefore, y-intercept is (0, 0)

From the graph of the data, created with MS Excel, the y-intercept is 50 which is not equal to zero, therefore, the relationship between the data is not a proportional relationship

8. The given data is presented in a tabular form as follows;

\(\begin{array}{ccc}Number \ of \ Minutes&&Words \ Typed\ \\1&&50\\2&&90\\3&&140\\4&&180\end{array}\)

From the graph of the data, we have that the y-intercept of the line of best fir is 5, therefore, the relationship is not a proportional relationship

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Answer:

∠ GOF = 53°

Step-by-step explanation:

∠ LON and ∠ GOF are vertically opposite angles and are congruent, then

∠ GOF = 53°

When comparing more than two treatment means, using an ANOVA (Analysis of Variance) instead of several t-tests has several advantages:

1. Reduced Type I Error: The use of multiple t-tests increases the likelihood of making a Type I error, which is the rejection of a true null hypothesis. By contrast, ANOVA allows you to test all means simultaneously, which reduces the overall probability of making a Type I error.

2. Increased Power: ANOVA increases the power to detect significant differences among means by taking advantage of the variance within groups and between groups.

3.  Control for Experimental Error: ANOVA also accounts for experimental error and random variability, which helps to improve the accuracy of the results.

4. Better Interpretation of Results: ANOVA provides more information about the relationship between the variables being tested. It can indicate whether there is an overall significant difference between the groups, and if so, which specific groups differ significantly from each other.

In summary, ANOVA is a more comprehensive statistical tool that can help to reduce errors, increase power, and provide more informative results when comparing more than two treatment means.

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The equation has no solution for x and the formula that correctly solves for the other base, b is b = 2/9A - 35

The equation is given as:

b = 2A/h

There is no occurrence of x in the above equation

This means that the equation has no solution for x

Here, we have:

A = 9/2(b+35)

Multiply both sides by 2

2A = 9(b+35)

Divide both sides by 9

2/9A = b + 35

Subtract 35 from both sides

b = 2/9A - 35

Hence, the formula that correctly solves for the other base, b is b = 2/9A - 35

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No, we cannot conclude that the true mean age is 30.

To determine if the true mean age is 30, we need to perform a hypothesis test. Given that the population variance is known, we can use a one-sample z-test.

Null Hypothesis (H₀): The true mean age is 30.

Alternative Hypothesis (H₁): The true mean age is not 30.

We will set the significance level (α) at 0.05.

Calculate the standard error of the mean (SEM):

SEM = √(population variance / sample size) = √(20 / 10) = √2 ≈ 1.414

Calculate the test statistic (z-score):

z = (sample mean - hypothesized mean) / SEM = (27 - 30) / 1.414 ≈ -2.121

Determine the critical z-values based on the significance level (α/2 = 0.025 for a two-tailed test) using a z-table or calculator. In this case, for α = 0.05, the critical z-values are approximately ±1.96.

Compare the calculated z-score with the critical z-values:

Since |-2.121| > 1.96, we reject the null hypothesis.

Based on the hypothesis test, there is enough evidence to reject the claim that the true mean age is 30.

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To determine the cost of the cell phone plan given the number of minutes used, we can break it down into two scenarios: when the number of minutes is within the 500 free minutes, and when it exceeds the 500 free minutes.

If the number of minutes used is within the 500 free minutes:

In this case, the cost of the cell phone plan is only the basic charge of $35 per month.

If the number of minutes used exceeds the 500 free minutes:

In this case, the cost of the additional minutes is calculated at a rate of 10 cents per minute. Let's denote the number of additional minutes as x. The cost of the additional minutes can be represented as 0.10x.

Therefore, the total cost of the cell phone plan, including the basic charge and any additional minutes, can be expressed as:

Total cost = Basic charge + Cost of additional minutes

Given that the basic charge is $35, we can write:

Total cost = $35 + 0.10x

To summarize:

If the number of minutes used is within the 500 free minutes, the total cost is $35.

If the number of minutes used exceeds the 500 free minutes, the total cost is $35 + 0.10x.

Note: It's important to consider any additional charges or fees that may be applicable to the cell phone plan. The given information states the basic charge and the charge for additional minutes, but other factors such as taxes or surcharges may also affect the total cost.

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The 20th percentile is 20,

The 25th percentile is 22.50.

The 65th percentile is 28.

The 75th percentile is 29.

Given values:

27, 25, 20, 15, 30, 34, 28, and 25.

n = 8

sorting the data gives:

15, 20, 25, 25, 27, 28, 30, and 34.

= 20/100 * 8

= 1.6 ≈ 2

1.6 is rounded to 2, the second value is in the sorted data set is 20 hence the 20th percentile is 20

= 25/100 * 8

= 2

Since 2 is an integer, the mean of the 2nd and the 3rd values in the sorted data set gives the 25th percentile.

( 20 + 25 ) / 2 = 22.5

hence the 25th percentile is 22.50

= 65/100 * 8

= 5.2 ≈ 6

5.6 is rounded to 6, the sixth value is in the sorted data set is 28 hence the 65th percentile is 28

= 75/100 * 8

= 6

Since 6 is an integer, the mean of the 6th and the 7th values in the sorted data set gives the 75th percentile.

( 28 + 30 ) / 2 = 29

hence the 75th percentile is 29

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Answer:

2x+11+11+x+3+3x-2=59

6x+23=59

6x=36

x=6

3x-2=18-2=16

Step-by-step explanation:

Answer: PQ=16

Step-by-step explanation:

59-11-11-3+2=36

36/6x=6

3(6)-2=16

Answer:

11 is the GCF for both 22 and 55.

Step-by-step explanation:

I looked it up.

The correct population should she answered is  is A) 7th graders in the school

To find out the favorite after-school activity of 7th grade students, Ms. Smith should sample the population directly related to the question. In this case, the most appropriate population to sample would be:

A) 7th graders in the school

By sampling the 7th graders in the school, Ms. Smith can directly gather information about their favorite after-school activities. This population consists of the students themselves who engage in the activities, and they would have first-hand knowledge and experience of their preferences.

B) Parents of 7th graders in the school might provide some insights, but they may not accurately represent the activities chosen by the students themselves.

C) Other 7th-grade teachers in the school may have some observations, but it may not reflect the preferences of the entire 7th-grade student population.

D) Sampling all students in the school would be broader than necessary and may not specifically address the favorite after-school activities of 7th-grade students.

Therefore, option A) 7th graders in the school is the most suitable population for Ms. Smith to sample in order to answer her question about the favorite after-school activity of 7th-grade students.

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We cannot verify if each of the four functions g1(x), g2(x), g3(x), and g4(x) have a fixed point at p when f(p) = 0.

To verify that if f(p) = 0, then each of the four functions g1(x), g2(x), g3(x), and g4(x) have a fixed point at p, we need to substitute p into each function and check if the result is equal to p.

g1(x) = (3+x-2x^2)^(1/4)

Let's substitute p into g1(x):

g1(p) = (3+p-2p^2)^(1/4)

To verify if g1(p) = p, we need to show that (3+p-2p^2)^(1/4) = p.

Since this is not an algebraic manipulation that can be solved easily, we cannot confirm if g1(x) has a fixed point at p without further calculations or approximations.

g2(x) = (x+3-x^4/2)^(1/2)

Let's substitute p into g2(x):

g2(p) = (p+3-p^4/2)^(1/2)

To verify if g2(p) = p, we need to show that (p+3-p^4/2)^(1/2) = p.

Since this is not an algebraic manipulation that can be solved easily, we cannot confirm if g2(x) has a fixed point at p without further calculations or approximations.

g3(x) = (x+3/x^2+2)^(1/2)

Let's substitute p into g3(x):

g3(p) = (p+3/p^2+2)^(1/2)

To verify if g3(p) = p, we need to show that (p+3/p^2+2)^(1/2) = p.

Since this is not an algebraic manipulation that can be solved easily, we cannot confirm if g3(x) has a fixed point at p without further calculations or approximations.

g4(x) = (3x^4+2x^2+3)/(4x^3+4x-1)

Let's substitute p into g4(x):

g4(p) = (3p^4+2p^2+3)/(4p^3+4p-1)

To verify if g4(p) = p, we need to show that (3p^4+2p^2+3)/(4p^3+4p-1) = p.

Since this is not an algebraic manipulation that can be solved easily, we cannot confirm if g4(x) has a fixed point at p without further calculations or approximations.

Therefore, based on algebraic manipulations alone, we cannot verify if each of the four functions g1(x), g2(x), g3(x), and g4(x) have a fixed point at p when f(p) = 0. Further calculations or approximations would be required to determine the fixed points of these functions.

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We know that the time is inversely proportional to the work.

Let t₁ is the time taken by Jeff and t₂ is the time taken by Jeff's dad.

We know that the formula:

Given:

Therefore,

Hence, it took Jeff's dad 12hours to wash the car himself.

Answer: 3, 4, 5

Step-by-step explanation:

(1) False, those two angles added together would be equal to angle 1, not 4.

(2) False. Angles 2, 3, 5 added together would make 180, not 3, 4, 5.

(3) True, an interior angle and its corresponding exterior angle are supplementary.

(4) True by the exterior angle theorem.

(5) True because the angles in a triangle add to 180 degrees

Answer:

Step-by-step explanation:

Slope is expressed as shown;

m = y2-y1/x2-x1

Given

m = 1/3

Coordinates A(x,6) B(2,4)

Substitute and get x

1/3 = 4-6/2-x

1/3 = -2/2-x

Cross multiply

2-x = 3(-2)

2-x = -6

-x = -6-2

-x = -8

x = 8

Hence the value of x is 8

By using the concept of error, the result obtained is-

Actual rating of 3 percent margin of error on a 50 percent approval rating is as low as 50 = 47%

What is error?

The difference between the measured value and the actual value is called error. Analysis of error is a crucial task mainly in mathematics and physics. In an experiment of physics, error analysis is a very important part. Error is also very important in numerical analysis of mathematics.

Actual rating of 3 percent margin of error on a 50 percent approval rating = 50 % \(\pm\) 3 %

Actual rating of 3 percent margin of error on a 50 percent approval rating is as low as 50 - 3 = 47%

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Step-by-step explanation:

according to the table, the rate of change of the water level =

(40-50)/(2-0)

= -10/2

= -5 ft/hr

_________

\(\:\)

Time = 2 - 0 = 2

Water level = 40 - 50 = -10

Soo :

-10/2 = -5ft/hr

a. The probability of drawing a red ball from Box 1 is 30%.

b. If a red ball is drawn from Box 1, the probability of drawing another red ball from Box 2 on the second round is 60%.

c. If the first draw from Box 1 was black, the conditional probability of drawing a red ball from Box 3 on the second draw is 80%.

d. The probability of drawing two red balls at both draws, without any prior information, is 46%.

e. The probability of drawing a red ball at the first draw and a black ball at the second draw, without any prior information, is 21%.

a. The probability of drawing a red ball from Box 1 can be calculated by dividing the number of red balls in Box 1 by the total number of balls in Box 1. Therefore, the probability is 3/(3+7) = 3/10 = 0.3 or 30%.

b. Since a red ball was drawn from Box 1, we only consider the balls in Box 2. The probability of drawing a red ball from Box 2 is 6/(6+4) = 6/10 = 0.6 or 60%. Therefore, the probability of drawing another red ball when the first ball drawn from Box 1 is red is 60%.

c. If the first draw from Box 1 was black, we only consider the balls in Box 3. The probability of drawing a red ball from Box 3 is 8/(8+2) = 8/10 = 0.8 or 80%. Therefore, the conditional probability of drawing a red ball from Box 3 when the first ball drawn from Box 1 was black is 80%.

d. Before any draws, the probability of drawing two red balls at both draws can be calculated by multiplying the probabilities of drawing a red ball from Box 1 and a red ball from Box 2. Therefore, the probability is 0.3 * 0.6 = 0.18 or 18%. However, since there are two possible scenarios (drawing red balls from Box 1 and Box 2, or drawing red balls from Box 2 and Box 1), we double the probability to obtain 36%. Adding the individual probabilities of each scenario gives a total probability of 36% + 10% = 46%.

e. Before any draws, the probability of drawing a red ball at the first draw and a black ball at the second draw can be calculated by multiplying the probability of drawing a red ball from Box 1 and the probability of drawing a black ball from Box 2 or Box 3. The probability of drawing a red ball from Box 1 is 0.3, and the probability of drawing a black ball from Box 2 or Box 3 is (7/10) + (2/10) = 0.9. Therefore, the probability is 0.3 * 0.9 = 0.27 or 27%. However, since there are two possible scenarios (drawing a red ball from Box 1 and a black ball from Box 2 or drawing a red ball from Box 1 and a black ball from Box 3), we double the probability to obtain 54%. Adding the individual probabilities of each scenario gives a total probability of 54% + 10% = 64%.

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The perimeter of the triangle is 170 inches.

To solve for the perimeter, we first need to find the length of the perpendicular height. We can do this using the sine function:

sin(35°) = 46/x

x = 46/sin(35°) = 65 inches

Now that we know the length of the perpendicular height, we can find the length of the base of the triangle using the cosine function:

cos(35°) = 65/x

x = 65/cos(35°) = 75 inches

The perimeter of the triangle is the sum of the lengths of the three sides, so the perimeter is:

P = 65 + 75 + 30

= 170 inches

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Answer:

Right triangle

Step-by-step explanation:

it has one 90 degree angle

Answer:

Right Triangle

Step-by-step explanation:

109x is the expression that could be used to determine the total number of cupcakes made given that a parent made x cupcakes for each of the 109 students in the fourth grade. This can be obtained by forming the algebraic expression for the given conditions.

Given that,

A parent made x cupcakes for each of the 109 students in the fourth grade.

The total number of students in the class = 109

The number of cupcakes one student gets = 1 × x = x

⇒ The number of cupcakes 109 student gets = 109 × x = 109x

Hence 109x is the expression that could be used to determine the total number of cupcakes made given that a parent made x cupcakes for each of the 109 students in the fourth grade.

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For the given problem:

Let the number = x

Two less than the quotient of a number and 8 is 1/4

so, we can write the following expression:

Now, solve the equation to find (x)

Multiplying by (8) to eliminate the denominators:

Add (6) to both sides:

So, the answer will be the number is 18

The volume of the solid generated by revolving the region between y=2sin(x) and the x-axis for 6 ≤ x ≤ 5π/6 about the y-axis is 2π (6sin(6) - 5√3) cubic units.

To find the volume of the solid generated by revolving the region between y=2sin(x) and the x-axis for 6 ≤ x ≤ 5π/6 about the y-axis, we use the method of cylindrical shells.

The formula for the volume of the solid generated by revolving the region between y=f(x) and the x-axis for a ≤ x ≤ b about the y-axis is given by:

V = 2π ∫[a,b] x f(x) dx

In this case, f(x) = 2sin(x) and the interval is 6 ≤ x ≤ 5π/6, so we have:

V = 2π ∫[6,5π/6] x (2sin(x)) dx

Using integration by parts, we obtain:

V = -2π [x cos(x)]6^5π/6 + 2π ∫[6,5π/6] cos(x) dx

V = 2π [x sin(x)]6^5π/6 - 2π sin(5π/6) + 2π sin(6)

Simplifying the expression, we get:

V = 2π (6sin(6) - 5√3)

Therefore, the volume of the solid generated by revolving the region between y=2sin(x) and the x-axis for 6 ≤ x ≤ 5π/6 about the y-axis is 2π (6sin(6) - 5√3) cubic units.

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