Describe what happens to the graph of y=a(x-2)^2+5 as ‘a’ varies

To find the lengths of the sides of the triangle with vertices at a(0, -4), b(0, -9), and c(-2, -5), we can use the distance formula:

Distance = √((x2 - x1)² + (y2 - y1)²)

Let's label the sides of the triangle as AB, BC, and CA, where A = (0, -4), B = (0, -9), and C = (-2, -5).

Length of AB:

AB = √((y2 - y1)² + (x2 - x1)²) (Note: We switched the order of x and y for this calculation to match the order of the points given)

= √((-9 - (-4))² + (0 - 0)²)

= √(5²)

= 5

Length of BC:

BC = √((y2 - y1)² + (x2 - x1)²)

= √((-5 - (-9))² + (-2 - 0)²)

= √(4² + 4²)

= √32

= 4√2

= 5.66

Length of CA:

CA = √((y2 - y1)² + (x2 - x1)²)

= √((-4 - (-5))² + (0 - (-2))²)

= √(1² + 2²)

= √5

= 2.24

Therefore, the lengths of the sides of the triangle areAB = 5

BC = 4√2 ≈ 5.66

CA = √5 ≈ 2.24

Since they are unequal, the triangle is a scalene triangle.

2)
To determine if AABC is a right-angled triangle, we need to check if any of its angles measure 90 degrees. We can use the slopes of the sides to determine this.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

Using this formula, we can calculate the slopes of the three sides of AABC:

Slope of AB:

AB passes through the points A(0, -4) and B(0, -9).

slope_AB = (-9 - (-4)) / (0 - 0) = -5/0 (undefined)

Note that the slope of AB is undefined, since the line is vertical and has no defined slope.

Slope of BC:

BC passes through the points B(0, -9) and C(-2, -5).

slope_BC = (-5 - (-9)) / (-2 - 0) = 2

Slope of CA:

CA passes through the points C(-2, -5) and A(0, -4).

slope_CA = (-4 - (-5)) / (0 - (-2)) = 1/2

Since AB is vertical and has undefined slope, we cannot use its slope to determine the angle at A. However, we can use the slopes of BC and CA to check the other angles.

The angle at B is opposite the side BC, so we need to check if the slope of BC is the negative reciprocal of the slope of AB. Since AB has undefined slope, we cannot determine its reciprocal or negative reciprocal. Therefore, we cannot use this method to check if the angle at B is 90 degrees.

The angle at C is opposite the side CA, so we need to check if the slope of CA is the negative reciprocal of the slope of BC. If it is, then the angle at C is 90 degrees. Let's check:

slope_BC * slope_CA = 2 * 1/2 = 1

The negative reciprocal of slope_BC is -1/2.

Since slope_CA is not equal to the negative reciprocal of slope_BC, we can conclude that the angle at C is not 90 degrees.

Therefore, we CANNOT conclude that AABC is a right-angled triangle based on the slopes of its sides.

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Answer:

[tex]-\frac{5}{3}[/tex]

Step-by-step explanation:

[tex]m (slope)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

[tex]m=\frac{-1-(-6)}{5-8}[/tex]

[tex]m=-\frac{5}{3}[/tex]

The distance between Patrick and Clarissa cannot be determined as only one variable has been given in the question. Not sufficient data provided.

Using congruency, length of the missing side of the triangle =- 8.5 units.

The dimensions of the sides and angles of two or more triangles determine whether they are congruent. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are equal, two triangles are said to be congruent. They are the exact same size and form. Triangles can satisfy a wide variety of congruence requirements.

In the 1st image:

The distance between Patrick and Clarissa cannot be determined as only one variable has been given in the question. Not sufficient data provided.

In the 2nd image:

Let x be the missing length of the triangle.

As the ray is an angle bisector,

Due to congruency:

11.9/7 = x/5

⇒ (11.9 × 5)/7 = x

⇒ x = 8.5 units.

Therefore, length of the missing side of the triangle =- 8.5 units.

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Answer:2

Step-by-step explanation:

When raising an exponential expression to a power, you need to multiply the exponents. Applying this rule to (y^3)^5, we get (y^3)^5 = y^(3*5) = y^15.

So, (y^3)^5 • y^4 = y^15 • y^4 = y^(15+4) = y^19.

Therefore, the simplified expression is y^19, and the answer is 2.

a. The product of the expressions is 15[tex]x^{3}[/tex] - 26[tex]x^{2}[/tex] + 26x - 24.

b. No, the product of [3x - 4 and 5[tex]x^{2}[/tex] - 2x + 6] and [4 - 3x and 5[tex]x^{2}[/tex] - 2x + 6] is not same.

The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

All real numbers are thought to be sufficiently explained by the four basic operations, also referred to as "arithmetic operations." In mathematics, quotient, product, sum, and difference are the operations that come after division, multiplication, addition, and subtraction.

a. We are given two expressions as 3x - 4 and 5[tex]x^{2}[/tex] - 2x + 6.

Using the multiplication operation, we get the product as

⇒ (3x - 4) * (5[tex]x^{2}[/tex] - 2x + 6)

⇒ 15[tex]x^{3}[/tex] - 6[tex]x^{2}[/tex] + 18x - 20[tex]x^{2}[/tex] + 8x - 24

⇒ 15[tex]x^{3}[/tex] - 26[tex]x^{2}[/tex] + 26x - 24

b. The product of 4 - 3x and 5[tex]x^{2}[/tex] - 2x + 6 is as follows,

⇒ (4 - 3x) * (5[tex]x^{2}[/tex] - 2x + 6)

⇒ 20[tex]x^{2}[/tex] - 8x + 24 - 15[tex]x^{3}[/tex] + 6[tex]x^{2}[/tex] - 18x

⇒ 26[tex]x^{2}[/tex] - 15[tex]x^{3}[/tex] - 26x + 24

This is not same as the product of previous expressions.

Hence, the required solutions have been obtained.

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Answer:

Step-by-step explanation the answer will be 20 each

:

The triangle formed by a complex number and its complex conjugate with the origin is an isosceles right triangle.

When a complex number and its complex conjugate are plotted in the complex plane, the two points are reflected about the real axis. Thus, the triangle formed by these points and the origin is an isosceles triangle with the real axis as its axis of symmetry.

Since the complex conjugate has the same magnitude as the original complex number, the two sides of the isosceles triangle have equal length. Therefore, the angle opposite the base (i.e., the origin) is a right angle, since it is the angle between the real axis and the imaginary axis.

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Step-by-step explanation:

The correct statement is: The amplitude will decrease, and the sound will become quieter.

When the compressions in a sound wave are very close together, the amplitude of the wave decreases. Amplitude is the measure of the maximum displacement of the particles of the medium from their rest position. As the amplitude of the wave decreases, the energy supplied by the vibrating source decreases, and the sound becomes quieter.

The dialatiοn pοint οf are:5) quadrilateral JKLM after a dilatiοn with a scale factοr οf 1/4 cantered at pοint K is J(7, 12), K(8, 12), L(7, 9), and M(4, 10).6) triangle XYZ after the dilatiοn is X'Y'Z', where X'(0, -3), Y'(-6, -9), and Z'(-12, 0).

A scale factοr is defined as the ratiο between the scale οf a given οriginal οbject and a new οbject, which is its representatiοn but οf a different size (bigger οr smaller).

5).Tο find the cοοrdinates οf the image οf quadrilateral JKLM after a dilatiοn with a scale factοr οf 1/4 centered at pοint K, we need tο perfοrm the fοllοwing steps:

Find the cοοrdinates οf pοint K after dilatiοn.

Since the center οf dilatiοn is K, the image οf K will be itself.

Therefοre, the cοοrdinates οf the image οf pοint K will be (8, 12).

Find the cοοrdinates οf the images οf pοints J, L, and M after dilatiοn.

Tο find the image οf pοint J, we can use the fοrmula fοr dilatiοn:

(image οf J) = (center οf dilatiοn) + (scale factοr) x (vectοr frοm center tο J)

(image οf J) = (8, 12) + (1/4)(vectοr JO)(image οf J) = (8, 12) + (1/4)(-4, 0)

(image οf J) = (7, 12)Similarly, we can find the images οf pοints L and M:

(image οf L) = (8, 12) + (1/4)(vectοr ZK)(image οf L) = (8, 12) + (1/4)(-4, -12)

(image οf L) = (7, 9)(image οf M) = (8, 12) + (1/4)(vectοr KM)(image οf M) =

(8, 12) + (1/4)(-16, -8)(image οf M) = (4, 10)

Therefοre, the image οf quadrilateral JKLM after a dilatiοn with a scale factοr οf 1/4 centered at pοint K is J(7, 12), K(8, 12), L(7, 9), and M(4, 10).

6). Tο find the image οf triangle XYZ after a dilatiοn centered at the οrigin with a scale factοr οf k = -3/2, we need tο apply the dilatiοn fοrmula tο each vertex.

Fοr vertex X(0, 2):

x-cοοrdinate οf image = k * x-cοοrdinate οf X = -3/2 * 0 = 0y-cοοrdinate οf image = k * y-cοοrdinate οf X = -3/2 * 2 = -3

Sο the image οf vertex X is X'(0, -3).

Fοr vertex Y(4, 6):x-cοοrdinate οf image = k * x-cοοrdinate οf Y = -3/2 * 4 = -6y-cοοrdinate οf image = k * y-cοοrdinate οf Y = -3/2 * 6 = -9

Sο the image οf vertex Y is Y'(-6, -9).

Fοr vertex Z(8, 0):x-cοοrdinate οf image = k * x-cοοrdinate οf Z = -3/2 * 8 = -12y-cοοrdinate οf image = k * y-cοοrdinate οf Z = -3/2 * 0 = 0

Sο the image οf vertex Z is Z'(-12, 0).

Therefοre, the image οf triangle XYZ after the dilatiοn is X'Y'Z', where X'(0, -3), Y'(-6, -9), and Z'(-12, 0).

Hence, the dialatiοn pοint οf are:5) quadrilateral JKLM after a dilatiοn with a scale factοr οf 1/4 centered at pοint K is J(7, 12), K(8, 12), L(7, 9), and M(4, 10).6) triangle XYZ after the dilatiοn is X'Y'Z', where X'(0, -3), Y'(-6, -9), and Z'(-12, 0).

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Therefore, the surface area of the prism is 139 square feet.

Area is the measure of the size of a two-dimensional region or surface, typically measured in square units such as square inches or square meters. It is calculated by multiplying the length of a shape by its width or height, or by using specific formulas depending on the shape. The concept of area is important in various fields such as mathematics, geometry, physics, engineering, and more.

Here,

To calculate the surface area of the prism, we need to find the area of each face and add them up.

The prism has two congruent triangular faces and three rectangular faces. The area of each triangular face is 1/2(base x height), where the base is 6 feet and the height is 7 feet, so the area of each triangular face is:

1/2(6)(7) = 21 square feet

The rectangular faces have dimensions of 6 feet by 5 feet, 7 feet by 5 feet, and 7 feet by 6 feet. The areas of these faces are:

6 x 5 = 30 square feet

7 x 5 = 35 square feet

7 x 6 = 42 square feet

Adding up the areas of all five faces, we get:

2(21) + 30 + 35 + 42 = 139 square feet

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Complete question:

A prism 5 feet tall whose base is a right triangle with legs length 6 feet and 7 feet. Find the area of the combined figure.

Waymon's hourly wage for the first 40 hours of labour was thus S9.68. equation

An equation in mathematics is a declaration of the equality of two expressions. Two sides of an equation are divided using the algebraic symbol (=). As an example, the assertion "2x + 3 = 9" is supported by the argument that it is true. Finding the value or values of the variable(s) needed to make the equation true is the aim of equation solving. It is possible to create regular or nonlinear, straightforward or complex equations that include one or more elements.

Let's name Waymon's usual hourly wage for the first 40 hours of labour "x". Based on the facts provided, we can then construct the following equation:

[tex]40x + 3(x + 1.5x) = 479.27[/tex]

Waymon's normal compensation for 40 hours of work is represented by the first term, while his overtime pay for 3 hours of work at time-and-a-half is represented by the second term (or 1.5 times his regular rate).

When we simplify this equation, we get:

[tex]40x + 4.5x = 479.27 - 48.4744.5x = 430.80 \\x = 9.68[/tex]

Waymon's hourly wage for the first 40 hours of labour was thus S9.68.

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The value of f^−1(4)f is the result of applying the inverse of the function f to 4.

To calculate f^−1(4)f, the inverse of the function f must be applied to 4.

Firstly, the inverse function of f, denoted as f^−1, is obtained by exchanging the x and y coordinates of all points on the graph of f. This means that the equation of f^−1 is obtained by solving the equation of f for y.

Next, 4 is substituted into the equation of f^−1 and the resulting equation is solved for x. This gives the value of f^−1(4).

Finally, the value of f^−1(4) is then substituted into the equation of f and the resulting equation is solved for y. This gives the value of f^−1(4)f.

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The formula for the Pythagorean theorem in 3 dimensions is [tex]d^2 = a^2 + b^2 + c^2.[/tex]

What is Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be extended to 3 dimensions using the same principle.

In the case of a cube with base a, width b, and height c, we can imagine a right triangle that is formed by one of the diagonals of the cube and one of the faces. Let d be the length of this diagonal.

By the Pythagorean theorem in 3 dimensions, we have:

[tex]d^2 = a^2 + b^2 + c^2[/tex]

To see why this is the case, imagine that we have a right triangle in three dimensions, with sides of length a, b, and c.

The hypotenuse of this triangle is the diagonal of a rectangular box with sides of length a, b, and c. By the Pythagorean theorem, the square of the length of the diagonal is equal to the sum of the squares of the lengths of the other two sides, which gives us the formula above.

Therefore, the formula for the Pythagorean theorem in 3 dimensions is [tex]d^2 = a^2 + b^2 + c^2.[/tex]

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The answer is ,the statement  that are true are: A. The small popcorn has  volume of about 25.13 cubic inches , C. The volume of two small popcorn is less than  volume of one large popcorn , E. The volume of  large popcorn is about 34.68 cubic inches greater than  volume of small popcorn (135.09 - 25.13 = 109.96).

A cone is  three-dimensional geometric shape that tapers smoothly from  flat, usually circular base to  point called the apex or vertex. The base of  cone can be any closed shape, but is usually a circle. A cone can be right or oblique, depending on whether or not the axis passing through the apex and the center of the base is perpendicular to the base.

The formula for volume of  cone are V = (1/3)πr^2h, where r is radius and h is height.

For the small popcorn cone:

radius =diameter/2 =4/2=2 inches

V = (1/3)π(2²)(6) ≈ 25.13 cubic inches

For the large popcorn cone:

radius = diameter/2 = 5/2 = 2.5 inches

V = (1/3)π(2.5²)(9) ≈ 59.81 cubic inches

Therefore, the statements that are true are:

A. The small popcorn has  volume of about 25.13 cubic inches

C. The volume of two small popcorn is less than volume of one large popcorn

E. The volume of  large popcorn is about 34.68 cubic inches greater than  volume of small popcorn (135.09 - 25.13 = 109.96)

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In this case, the graph crosses the x-axis that is x intercept at x = -3 and x = 3,thus the answer is B. (x-3)(x+3).

Intercept is the point where a graph intersects with either the x-axis or y-axis. It is the value at which a line crosses the axis.

To determine the factored form of the related polynomial, we must first identify the zeroes of the function.

A zero is a point on the graph where the value of the function is equal to zero.

In the graph given, there are two zeroes, one at x=-3 and one at x=3.

The factored form of the polynomial can be expressed as (x-3)(x+3).

This is because the two zeroes of the function correspond to the two linear factors,

x-3 and x+3.

This matches the equation of the polynomial given in the graph, confirming that the correct factored form is (x-3)(x+3).

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Answer:

Marcus has to multiply the height times the length times the width.

Step-by-step explanation:

The formula for the volume of a rectangular prism is the following:

Volume of rectangular prism = length x width x height

Answer:

36

Step-by-step explanation:

f(4) = (4+2)² = (6)² = 36

Answer: 2

Step-by-step explanation:

7x-5 =3x+9

x = 4

7x4-5= 23

3x4+9=21

∠PQT=∠PQS+∠SQT

23=21+∠SQT

∠SQT=2

Answer:

Step-by-step explanation:

The triangles are similar so the ratio of their corresponding sides will be the same. That is, TV/RV is equal to TS/RQ. This gives us the equation 12/x+6=16/x+1. Cross multiplying, this becomes 12x+12=16x+96. Simplifying, we get x=21. Therefore, RV=12+x-6=x+6=21+6=27.

Answer:

TV/RV=TS/RQ, 12/x+6=16/x+1, 12x+12=16x+96, x=21. Therefore, RV=12+x-6=x+6=21+6=27.

Step-by-step explanation:

Answer:

rotation to the right in the coordinate plane

The BEST definition for clockwise rotation on the coordinate plane is "rotation to the right in the coordinate plane." option B is correct.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.

"Rotation to the right in the coordinate plane" is the BEST definition for clockwise rotation on the coordinate plane. In the coordinate plane, a rotation that turns a point or object to the right of the origin is referred to as being clockwise.

It follows that if a straight line were to stretch from the origin to the point or item, it would spin as it traveled in a clockwise direction. In comparison, a rotation in the other direction would cause the line to spin to the left.

Although notation can be used to depict the magnitude and direction of rotation in the coordinate plane, this does not always mean that the rotation is clockwise.

Therefore, Rotation to the right in the coordinate plane is the BEST definition for clockwise rotation on the coordinate plane.

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Answer:

4.275 L

Step-by-step explanation:

Pitcher starts with 6.2 L.

7 glasses

each glass has 0.275 L

total water in 7 glasses: 7 × 0.275 L = 1.925 L

water remaining in pitcher: 6.2 L - 1.925 L = 4.275 L

Answer: 4.275 L is left in the pitcher.

The equation of parabola for the given equation through which it satisfied the relation is     [tex]( \frac{1}{30} )x^2 + (\frac{1}{30} )x - 1[/tex].

What about parabola?

A parabola is a U-shaped curve that is formed by the graph of a quadratic function of the form y = [tex]ax^2 + bx + c[/tex]. The parabola is symmetric around its axis of symmetry, which is a vertical line passing through the vertex of the parabola. The vertex is the point where the parabola changes direction, either from increasing to decreasing or from decreasing to increasing. The direction of the opening of the parabola (upward or downward) is determined by the sign of the coefficient a in the quadratic equation. Parabolas are used in various fields, such as physics, engineering, and mathematics, to model various real-life phenomena.

Define equation:

An equation is a statement that shows the equality of two expressions, typically separated by an equals sign (=). It represents a mathematical relationship between two or more variables or constants, and is used to solve problems and make predictions.

In mathematics, equations can take various forms, including linear equations, quadratic equations, trigonometric equations, differential equations, and more. Depending on the type of equation, the variables may represent real or complex numbers, vectors, matrices, functions, or other mathematical objects.

According to the given information:

As, we know about the condition of the parabola,

Hence:

[tex]y = Ax^2 + Bx + C y-intercept: x=0 and y = -1\\\\ C = -1 y = Ax^2 + Bx - 1 (5,0) \\ \\0 = 25A + 5B - 1 (-6,0) \\\\ 0 = 36A - 6B - 1 1 \\\\ = 25A + 5B1 = 36A - 6B \\ \\ 0 = -11A - 11B 11B = -11A B = -A kX^2 - kx - 1 , k=A=-b (5,0)\\\\25k + 5k - 1 = 030k = 1k = 1/30 \\\\(-6,0)36k - 6k - 1 = 0\\\\30k = 1k=30 (\frac{1}{30} )x^2 + (\frac{1}{30} )x - 1 ,[/tex]

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Answer:

it's above

Step-by-step explanation:

yeah

The standard deviation for the given Poisson distribution is 0.8.

A Poisson distribution is used to model the number of events occurring in a fixed time interval when the events occur independently and at a constant rate. The mean and variance of a Poisson distribution are equal, and both are given by λ, the rate parameter.

Here, the given information is that the company receives an average of 0.64 purchase orders per minute. Therefore, λ = 0.64.

The formula for the variance of a Poisson distribution is σ² = λ, where σ is the standard deviation. Thus, substituting λ = 0.64 in this formula, we get σ² = 0.64. Taking the square root of both sides, we get σ = √0.64 = 0.8.

Therefore, the correct answer is 0.8.

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The correct statements are:

The median is six and one-fourth.

The mode cannot be determined from the given information.

The range is five and three-fourths.

The median measures the central tendency in a dataset, which is the middle value when the data is arranged in order from lowest to highest (or highest to lowest). If the dataset has an odd number of values, the median is the middle value. If there is an even number of values in the dataset, the median is the average of the two middle values.

The median is six and fourth: This statement is true based on the given information that there are 11 marks on the line plot above 6 and fourth, and 11 marks below.

The mode is seven and two-fourths: This statement is false based on the given information. The mode is the value that appears most frequently in the dataset, and there are no values that appear more than once in the given data.

The range is five and three-fourths: This statement is true based on the given information. The range is the difference between the highest and lowest values in the dataset, which in this case is 5 and three-fourths (8 minus 2 and one-fourth).

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Answer: -3

Step-by-step explanation: 3-3=0

Answer:

Step-by-step explanation:

bonsoir aider moi plus vite possible

3+?=0

3 + x = 0

x = -3

140 ml of Ensure
To prepare a tube feeding of Ensure 280 ml (50% solution), the nurse should use 140 ml of Ensure. Here's the explanation: Given: Ensure is available in a 240 ml can.

The required amount of Ensure is 280 ml (50% solution). Formula: To calculate the number of milliliters of Ensure to be used, we will use the formula: percent strength (as a decimal) × volume required ÷ percent strength in stock = volume to be administered Solution: Since Ensure is available in a 240 ml can, we will use that as our percentage strength in stock. Now we'll plug in the values:0.5 × 280 ÷ 1 = 140 ml Therefore, the nurse should use 140 ml of Ensure to prepare the feeding.

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Answer:

Step-by-step explanation:

Short answer? x is 52 degrees and y is 38 degrees.

But how did I get it you probably did not ask?

Let's run it back to the definitions of complementary and supplementary angles.

Complementary angles: 2 angles that add to get you a CLEAN 90 degrees

Supplementary angles: 2 angles that add to get you a CLEAN 180 degrees

So if x is supplementary to 128 degrees, it has to be 52 degrees to fit the definition. So yeah 180-128= 52 degrees

Now for x to be complementary to y, y has to 38 degrees. 90-52=38 degrees.

x: 52 degrees

y: 38 degrees

I teleport here again should you have anymore questions.

a)The function that describes the height of the rocket in terms of time t can be expressed as: h(t) = -16t² + 224t + 98

b) the maximum height of the rocket is 1378 feet at t=7sec.

c)the rocket will be more than 702 feet above ground level during the time interval [1.95, 18.55].

d)14.13 seconds will take it to hit the ground

The quadratic formula is a formula that gives the solutions (or roots) of a quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients, and x is the variable:

x = (-b ± √(b² - 4ac)) / (2a)

a) The function that describes the height of the rocket in terms of time t can be expressed as:

h(t) = -16t² + 224t + 98

where h(t) represents the height of the rocket (in feet) at time t (in seconds).

The -16t² term represents the effect of gravity, which causes the rocket's height to decrease over time. The 224t term represents the initial velocity of the rocket, which causes its height to increase over time. The constant term 98 represents the initial height of the rocket (at t = 0).

b) To determine the time at which the rocket reaches its maximum height, we need to find the vertex of the parabolic function h(t). The x-coordinate of the vertex is given by:

t = -b / (2a)

where a = -16 (the coefficient of t²) and b = 224 (the coefficient of t). Substituting these values, we get:

t = -224 / (2×(-16)) = 7

Therefore, the rocket reaches its maximum height at t = 7 seconds.

To find the maximum height, we can substitute t = 7 into the function h(t) and simplify:

h(7) = -16(7)²+ 224(7) + 98 = 1378

Therefore, the maximum height of the rocket is 1378 feet.

c) To find the time interval during which the rocket is more than 702 feet above ground level, we need to solve the inequality:

h(t) > 702

Substituting the expression for h(t), we get:

-16t² + 224t + 98 > 702

Simplifying and rearranging terms, we get:

16t² - 224t - 604 < 0

Using the quadratic formula, we can solve for the roots of the equation:

t = [224 ± √(224²- 4(16)(-604))] / (2(16))

t ≈ 1.95 or t ≈ 18.55

Therefore, the rocket will be more than 702 feet above ground level during the time interval [1.95, 18.55].

d) To find the time at which the rocket hits the ground, we need to find the root(s) of the function h(t) where h(t) = 0. This occurs when the rocket returns to ground level (i.e., h(t) = 0) after reaching its maximum height.

Substituting the expression for h(t), we get:

-16t²+ 224t + 98 = 0

Using the quadratic formula, we can solve for the roots of the equation:

t = [224 ± √(224² - 4(-16)(98))] / (2(-16))

t ≈ 14.13 or t ≈ -0.38

the rocket hits the ground after approximately 14.13 seconds.

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