What are the explanation sentences of this question

The value of cos (A+B) in simplest form is -136/305.

The standard formula for the cosine of the sum of two angles is:

cos(A+B) = cos A cos B - sin A sin B

We are given that cos A = 11/61 and sin B = 3/5. We can use this information to find the values of sin A and cos B by using the Pythagorean identity:

sin^2 A +[tex]cos^2[/tex]A = 1  =>  sin A = [tex]\sqrt(1 - cos^2[/tex] A)

cos^2 B + [tex]sin^2[/tex] B = 1 => cos B = [tex]\sqrt(1 - sin^2 B)[/tex]

Substituting the given values, we get:

sin A =[tex]\sqrt(1 - (11/61)^2) ~~ 60/61[/tex]

cos B = [tex]\sqrt(1 - (3/5)^2) = 4/5[/tex]

Now we can use these values to find the cosine of the sum of the angles:

cos(A+B) = cos A cos B - sin A sin B

= (11/61)(4/5) - (60/61)(3/5)

= (44/305) - (36/61)

= (44 - 180)/305

= -136/305

Note that since A and B are positive acute angles, A+B is also an acute angle, which means that its cosine is negative.

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Step-by-step explanation:

try this option (see the attached picture); the required point is marked with red colour.

The measures of the angles are ∠PQR = 96° and ∠PSR = 84°

Given is a cyclic quadrilateral, with angles ∠PQR = (7x - 2)° and ∠PSR = (5x+14)°.

We need to find the measure of the angles PQR and PSR,

So,

We know that the cyclic quadrilaterals have their opposite angles supplementary,

So,

∠PQR + ∠PSR = 180°

7x - 2 + 5x + 14 = 180°

12x + 12 = 180°

12x = 168°

x = 14°

Put the value of x in the angles, we get,

∠PQR = (7x - 2)°

∠PQR = 7 × 14 - 2

∠PQR = 96°

And,

∠PSR = 5 × 14 - 2

∠PSR = 84°

Hence the measures of the angles are ∠PQR = 96° and ∠PSR = 84°

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G = 14 - D/30 is the equation that is relating G to D

Given the information, we can create an equation that describes the amount of gas G Jane has in her car after she has traveled a distance D.

Since Jane's car uses one gallon of gas every 30 miles, for every mile she travels, she uses 1/30 of a gallon.

So, after traveling D miles, Jane has used D/30 gallons of gas.

Because she started with 14 gallons, we can subtract the amount used from the initial amount to find the remaining amount of gas G:

G = 14 - D/30

So, G represents the gallons of gas remaining in Jane's car after she has traveled D miles. This equation assumes that Jane doesn't refill her gas tank during her trip.

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a) The binomial probability distribution is solved

b) The mean is 3 and standard deviation is 1.2247

c) The mean is 3 and standard deviation is 1.2247

Given data ,

a)

To construct a binomial probability distribution, we can use the binomial probability formula:

P ( x ) = [ n! / ( n - x )! x! ] pˣqⁿ⁻ˣ

n = 5

p = 0.5

For x = 0:

P(X = 0) = (6C0) * (0.5^0) * ((1 - 0.5)^(6 - 0)) = 1 * 1 * 0.015625 = 0.015625

For x = 1:

P(X = 1) = (6C1) * (0.5)¹ * ((1 - 0.5)⁽⁶⁻¹⁾) = 6 * 0.015625 = 0.09375

For x = 2:

P(X = 2) = (6C2) * (0.5^2) * ((1 - 0.5)^(6 - 2)) = 15 * 0.015625 = 0.234375

For x = 3:

P(X = 3) = (6C3) * (0.5^3) * ((1 - 0.5)^(6 - 3)) = 20 * 0.015625 = 0.3125

For x = 4:

P(X = 4) = (6C4) * (0.5^4) * ((1 - 0.5)^(6 - 4)) = 15 * 0.015625 = 0.234375

For x = 5:

P(X = 5) = (6C5) * (0.5^5) * ((1 - 0.5)^(6 - 5)) = 6 * 0.015625 = 0.09375

For x = 6:

P(X = 6) = (6C6) * (0.5^6) * ((1 - 0.5)^(6 - 6)) = 1 * 0.015625 = 0.015625

Thus, the binomial probability distribution for the given table is as follows:

x: { 0, 1, 2, 3, 4, 5, 6 }

y: { 0.015625, 0.09375, 0.234375, 0.3125, 0.234375, 0.09375, 0.015625 }

b)

The mean is μ = Σ [ x P ( x ) ]

μ = (0 * 0.015625) + (1 * 0.09375) + (2 * 0.234375) + (3 * 0.3125) + (4 * 0.234375) + (5 * 0.09375) + (6 * 0.015625)

μ = 0 + 0.09375 + 0.46875 + 0.9375 + 0.9375 + 0.46875 + 0.09375

μ = 3

The standard deviation σ = √Σ (x²P(x) - μ ²)

Σ(x^2 * P(x)) = (0^2 * 0.015625) + (1^2 * 0.09375) + (2^2 * 0.234375) + (3^2 * 0.3125) + (4^2 * 0.234375) + (5^2 * 0.09375) + (6^2 * 0.015625)

= 0 + 0.09375 + 0.9375 + 2.8125 + 3.75 + 2.34375 + 0.65625

= 10.59375

Now, we can calculate the standard deviation:

σ = √(Σ(x^2 * P(x)) - μ^2)

σ = √(10.59375 - 3^2)

σ = √(10.59375 - 9)

σ = √(1.59375)

σ ≈ 1.261

Therefore, the standard deviation (σ) is approximately 1.261.

c)

To find the mean (μ) and standard deviation (σ) of a binomial probability distribution, we can use the following formulas:

Mean (μ) = n * p

Standard Deviation (σ) = sqrt(n * p * (1 - p))

Given the values n = 6 and p = 0.5, we can calculate the mean and standard deviation as follows:

Mean (μ) = 6 * 0.5 = 3

Standard Deviation (σ) = sqrt(6 * 0.5 * (1 - 0.5)) = sqrt(1.5) ≈ 1.2247

Hence , the mean of the binomial probability distribution is 3 and the standard deviation is 1.2247.

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Let's represent the height of the tree as "h". According to the problem, the wire supporting the tree is 5 m longer than the height of the tree, so the length of the wire is "h + 5". The wire is anchored at a point whose distance from the base of the tree is 35 m shorter than the height of the tree, so this distance is "h - 35".

We can now use the Pythagorean theorem to find the height of the tree. 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.

In this case, we have a right triangle formed by the tree, the ground and the wire. The height of the tree is one side of this triangle, and its length is "h". The distance from the base of the tree to where the wire is anchored is another side of this triangle, and its length is "h - 35". The wire itself is the hypotenuse of this triangle, and its length is "h + 5".

Applying the Pythagorean theorem, we have:

(h + 5)² = h² + (h - 35)²

Expanding and simplifying this equation:

h² + 10h + 25 = h² + h² - 70h + 1225

2h² - 80h + 1200 = 0

Dividing by 2:

h² - 40h + 600 = 0

We can solve this quadratic equation using factoring or by applying the quadratic formula. Factoring gives us:

(h - 30)(h - 20) = 0

So, h = 30 or h = 20.

Since both solutions are positive, either one could be a valid answer for this problem. However, if we have additional information or constraints that would allow us to choose one solution over the other, we could determine which one is correct. Without such information, we can only say that there are two possible answers: either h = 30 or h = 20.

En un número decimal, la parte que está antes del punto decimal se conoce como la parte entera del número, mientras que la parte que está después del punto decimal se conoce como la parte decimal.

La parte entera representa la cantidad completa de unidades o números enteros que se encuentran en el número. Por ejemplo, en el número decimal 12.345, la parte entera es 12, lo que significa que hay 12 unidades completas.

La parte decimal representa una fracción o porción del número que es menor que uno. En el ejemplo anterior, la parte decimal es .345, lo que significa que es una fracción de una unidad. La parte decimal se puede expresar en forma de fracción o como un número decimal.

La relación entre la parte entera y la parte decimal es que, juntas, representan el número decimal completo. Por ejemplo, el número decimal 12.345 se compone de la parte entera 12 y la parte decimal .345. La parte entera siempre se coloca a la izquierda del punto decimal, mientras que la parte decimal se coloca a la derecha del punto decimal.

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The value of x is 17

To determine the value, we need to know that;

Then, from the information given, we have that;

2x and 3x + 5 are complementary

Then, we have that;

2x + 3x + 5 = 90

collect the like terms, we have;

5x = 85

Divide by the coefficient of x, we get;

x = 17

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A thin, red-haired baby wearing a bonnet is the oldest of the group can be Baby 4

We are given that hospital maternity ward there are only five babies.

Baby 1 = heavier, with red hair.

Baby 2 = male and thin, with the same colour hair as Baby 3

Baby 3 = blonde and wears a bonnet.

Baby 4 = one of the three female babies

Baby 5 = wears sunglasses and is heavier.

Here All babies wear bonnets and at least one of the females wears sunglasses.

Because the baby we are finding is thin and red-haired so takes out Baby 1 because he is heavier.

This takes out Baby 2 because he has blonde hair.

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The correct reason for the missing part of the proof in order to prove is,

⇒ Corresponding angles are congruent.

We have to given that;

Consider the figure below with DE || BC and ∠1 ≅ ∠2.

Now, We know that;

The pairs of angles formed on the same side of the transversal that are either both obtuse or both acute and are called corresponding angles and are equal in size.

Hence, We get;

The correct reason for the missing part of the proof in order to prove is,

⇒ Corresponding angles are congruent.

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The volume of the cylinder is 769.3 cubic feet and Volume of rectangular pyramid is 432 cubic meter

The cylinder has a height of 5 ft and radius of 7 ft

We have to find the volume of the cylinder

Volume of cylinder =πr²h

Plug in the value of r and h in the formula

Volume of cylinder =3.14×7²×5

=3.14×49×5

=769.3 cubic feet

Volume of rectangular pyramid is 1/3(length ×width×height)

Volume =1/3(12×12×9)

=1296/3

=432 cubic meter

Hence, the volume of the cylinder is 769.3 cubic feet and Volume of rectangular pyramid is 432 cubic meter

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Answer:

Surface area = 298 ft^2

Step-by-step explanation:

One of the formula we can use for surface area of a rectangular box is given by:

SA = 2lw + 2lh + 2wh, where

In the figure, the height is 7 ft, the length is 11 ft, and the width is 4ft.  Thus, we plug in 7 for h, 11 for l, and 4 for w in the surface area formula to find the surface area of the rectangular box:

SA = 2(11 * 4) + 2(11 * 7) + 2(4 * 7)

SA = 2(44) + 2(77) + 2(28)

SA = 88 + 154 + 56

SA = 242 + 56

SA = 298

Thus, the surface area of the rectangular box is 298 square ft or 298 ft^2

Answer:

A. Add Equation A and Equation B.

Step-by-step explanation:

When you add Equation A and Equation B, you'd get (when combining like terms)

(7y + 3y) + (-4x + 4x) = (5 + 25), which becomes 10y = 30

This show us that adding the equations allows us to cancel (eliminate) the x variable.  Then we'd solve for y and later we'd solve for x by plugging in the value for y into either equation A or equation B.

n³a⁹= (9/64)r³ this equation shows that 125ncubica(a³) = 9 cubic r, as desired by cone and sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.

Since we have n ice spheres of radius a

the total volume of the ice spheres is given by V_ice = n × (4/3)πa³.

The volume of a cone is given by the formula V_cone = (1/3)πr²h

where r is the radius of the base and h is the height of the cone.

The slant height of the cone is also given as r.

Using the Pythagorean theorem, we can find the height of the cone, h, in terms of r:

h =√(r² - a²)

Substituting this value of h into the formula for the volume of the cone, we get:

V_cone = (1/3)πr² √(r² - a²)

V_ice = V_cone

n(4/3)πa³= (1/3)πr²√(r² - a²)

Simplifying this equation, we can cancel out the common factors of (1/3)π:

4na³= r²√(r² - a²)

Now, let's cube both sides of the equation to eliminate the square root:

(4na³)³ = (r²√(r² - a²))³

64n³a⁹ = r⁶ (r² - a²)

n³a⁹= (9/64)r³

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Answer:

Step-by-step explanation: your answer is b i had this quetion before

The volumes of the shapes are:

V = 90π in³, V = 1875.78π ft³, V = 240 cm³, V =500π cm³, V = 30 cm³.

Here, we have,

1.) Volume of cylinder

V= π×r²×h

Where

π=3.14

r=3

h=10in

V=3.14×3²×10

V = 90π in³

2.) Volume of cylinder

V=  π×r²×h

Where

π=3.14

r=17.5/2 ft

h=24.5 ft

V = 1875.78π ft³

3.) Volume of prism

V = l×w×h

so, we get,

l = 6cm, w = 8cm, h = 5cm

V = 240 cm³

4.)Volume of cylinder

V= π×r²×h

Where

π=3.14

r=20/2 cm

h=5 cm

V =500π cm³

5.) Volume of rectangular box

V = l×w×h

so, we get,

l = 2cm, w = 5cm, h = 3cm

V = 30 cm³

Hence, The solutions are: the volumes of the shapes are:

V = 90π in³, V = 1875.78π ft³, V = 240 cm³, V =500π cm³, V = 30 cm³.

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They all will play together when Ricky completes the work at 5:30 PM.

From the information given, we have;

Time which Ricky takes to complete the work

= 5 hrs = 5 x 60 = 300 mins

Ricky 's time allocation will be - 210 mins + 30 mins break + 90 mins post 4 = 5:30 he finishes the work.

Time which Jane  takes to complete the work = 90 × (7/3) = 210 mins

Jane 's time allocation will be - 210 mins  = 3:30 she finishes the work.

Time which Ramya takes to complete the work = 210 × (4/3) = 280 mins

Ricky's time allocation will be - 210 mins + 30 mins break + 70 mins post 4 = 5:10 she finishes the work.

Rajesh - 5:30 PM

Jane - 3:30 PM

Ann - 5:10 PM

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The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

We have,

To find the probability that an antigen test comes back positive, we need to consider both the true positive rate (probability of a positive test given that the person is infected) and the false positive rate.

Now,

Prevalence of coronavirus in California: 3 out of 1000

False positive rate of the antigen test: 0.05 (5 out of 100)

Let's calculate the probability of a positive test result.

The true positive rate can be calculated as 1 minus the false negative rate (probability of a negative test given that the person is infected):

True positive rate = 1 - 0.2 = 0.8 (or 80 out of 100)

The probability of a positive test result can be calculated using Bayes' theorem:

P(Positive test) = P(Positive test | Infected) x P(Infected) + P(Positive test | Not Infected) x P(Not Infected)

P(Positive test) = (0.8 x 3/1000) + (0.05 x 997/1000)

P(Positive test) = 0.0024 + 0.04985

P(Positive test) = 0.05225

Therefore,

The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

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Answer: The values of the six trigonometric functions for the angle formed by the positive x-axis and the line segment from (0,0) to (4,-3) are:

sinθ = -3/5

cosθ = 4/5

tanθ = -3/4

cscθ = -5/3

secθ = 5/4

cotθ = -4/3

Step-by-step explanation: To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle formed by the positive x-axis and a line segment, we need to determine the lengths of the sides of the right triangle formed.

In this case, the line segment goes from (0,0) to (4,-3). The horizontal length (adjacent side) is 4 units, and the vertical length (opposite side) is -3 units (since it goes downward).

Now, we can calculate the trigonometric functions:

Sine (sinθ) = opposite/hypotenuse = -3/5

Cosine (cosθ) = adjacent/hypotenuse = 4/5

Tangent (tanθ) = opposite/adjacent = -3/4

Cosecant (cscθ) = 1/sinθ = -5/3

Secant (secθ) = 1/cosθ = 5/4

Cotangent (cotθ) = 1/tanθ = -4/3

Therefore, the values of the six trigonometric functions for the angle formed by the positive x-axis and the line segment from (0,0) to (4,-3) are:

sinθ = -3/5

cosθ = 4/5

tanθ = -3/4

cscθ = -5/3

secθ = 5/4

cotθ = -4/3

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The value of arc CD in the intersecting chords is determined as 57⁰.

The value of arc CD is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

arc BCD = arc CD +  arc CB

arc BCD = 2 x 84⁰

arc BCD = 168⁰

168 = CD + CB

arc BAD = BA + AD

angle DCB = 180 - 84 ( opposite angles of a cyclic quadrilateral are supplementary)

angle DCB = 96⁰

arc BAD = 2 x 96 = 192⁰

192 = BA + AD

192 = 127 + AD

AD = 65⁰

The value of arc CD is calculated as follows;

Arc ADC = AD + CD

arc ADC = 2 x 61 = 122

122 = AD + CD

122 = 65 + CD

CD = 57⁰

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The correct answer is c. mass, height.

If you increase the mass of an object and raise it to a higher height, you will increase its amount of potential energy. Potential energy is directly proportional to both mass and height.

The potential energy of an object in a gravitational field is given by the equation:

Potential Energy = mass * gravitational acceleration * height

By increasing the mass (m) and the height (h), the potential energy (PE) of the object increases:

PE ∝ m * h

Therefore, increasing the mass and height of an object will increase its potential energy.

The solution is: : transformation maps the pre-image to the image is Rotation.

Here, we have,

A transformation of a triangle can be either dilation or reflection or rotation or a translation.

Dilation happens when a symmetric figure is formed with scale factor other than 1.  But here both triangles have same side length.  Hence no dilation

Reflection happens when it is reflected over a line as we see in a mirror. But here the two triangles are not looking as images on  a line.  Hence no reflection

Rotation is keeping the same shape but rotating through a certain angle.  Here DEF is rotated without disturbing its shape or size through a certain angle. Hence rotation is right

Translation is not right because there is no vertical or horizontal shift to get new triangle.

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complete question:

A transformation of ΔDEF results in ΔD'E'F'.

Which transformation maps the pre-image to the image?

The transformation is a dilation.

The transformation is a reflection.

The transformation is a rotation.

The transformation is a translation.

Answer: The equation x^n + y^n = (x + y)^n represents Fermat's Last Theorem for the case when the exponents are equal. According to Fermat's Last Theorem, there are no positive integer solutions (x, y, z) for the equation x^n + y^n = z^n, where n is a positive integer greater than 2.

However, in this case, we are looking for solutions where x ≠ y, so the equation x^n + y^n = (x + y)^n may have infinitely many positive integer solutions for certain values of n.

To find the values of n for which the equation has infinitely many positive integer solutions (x, y) with x ≠ y, we need to consider the equation x^n + y^n = (x + y)^n and see if there are any such values.

Let's analyze the equation for different values of n:

When n = 2:

When n = 3:

Therefore, when n = 3, the equation has infinitely many positive integer solutions (x, y) with x ≠ y.

When n > 3:

In summary, the equation x^n + y^n = (x + y)^n has infinitely many positive integer solutions (x, y) with x ≠ y when n = 3. For all other values of n greater than 1, there are no such solutions.

The solution to the system of equations is x = 4 and y = 8. The lines represented by the equations y = 2x and 6x - 2y = 8 Intersect at the point (4, 8).

To analyze the given equations:

Equation 1: y = 2x

Equation 2: 6x - 2y = 8

We can begin by examining Equation 1, which is in slope-intercept form (y = mx + b). In this equation, the coefficient of x is 2, representing the slope of the line. Therefore, the line described by Equation 1 has a slope of 2.

Now, let's move on to Equation 2. It can be rewritten by rearranging the terms:

6x - 2y = 8

-2y = -6x + 8

Dividing by -2 on both sides:

y = 3x - 4

By comparing Equation 2 with the slope-intercept form (y = mx + b), we can see that its slope is 3.

Comparing the slopes of the two equations, we observe that they are not equal. Since the slopes are different, the lines represented by the equations y = 2x and y = 3x - 4 are not parallel.

To determine if they intersect, we can equate the right-hand sides of the two equations:

2x = 3x - 4

By rearranging terms, we get:

x = 4

Substituting x = 4 back into Equation 1:

y = 2(4)

y = 8

Therefore, the solution to the system of equations is x = 4 and y = 8. The lines represented by the equations y = 2x and 6x - 2y = 8 intersect at the point (4, 8).

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A. Matrix AB is [tex]\left[\begin{array}{cc}24&37\\6&25\end{array}\right][/tex]

B. Matrix BA is [tex]\left[\begin{array}{cc}10&-12\\-1&39\end{array}\right][/tex]

C. No, in general, AB does not always equal BA.

A. To find AB we say [tex]\left[\begin{array}{cc}-3&5\\1&3\end{array}\right][/tex] × [tex]\left[\begin{array}{cc}-3&1\\3&8\end{array}\right][/tex]

which becomes [tex]\left[\begin{array}{cc}-3*-3 + 5*3& -3*3 + 5*8\\1*3 +3*3&1*1 + 8*3\end{array}\right][/tex] ⇒   [tex]\left[\begin{array}{cc}24&37\\6&25\end{array}\right][/tex]

B. To find Matrix BA  [tex]\left[\begin{array}{cc}-3&1\\3&8\end{array}\right][/tex] ×  [tex]\left[\begin{array}{cc}-3&5\\1&3\end{array}\right][/tex]

Which becomes [tex]\left[\begin{array}{cc}-3*3 + 1*1 &-3*5 + 1*3\\3*-3 + 8*1&3*5 + 8*3\end{array}\right][/tex] ⇒  [tex]\left[\begin{array}{cc}10&-12\\-1&39\end{array}\right][/tex]

C. For matrices A and B such that AB and BA both exist, AB will equal BA if and only if the matrices commute. A matrix commutes with another matrix if the order in which they are multiplied does not matter.

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The angle m∠BAC is 54 degrees.

If two chords intersect inside a circle, then the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Therefore, let's find the angle BAC.

Hence,

10x + 4 = 1 / 2 (12x + 15 + 9x - 12)

10x + 4 = 1 / 2 (21x + 3)

10x + 4 = 10.5x + 1.5

10x - 10.5x = 1.5 - 4

-0.5x = -2.5

divide both sides by -0.5

x = -2.5 / -0.5

x = 5

Therefore,

m∠BAC = 10(5) + 4

m∠BAC = 50 + 4

m∠BAC = 54 degrees

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Answer: The answer is (d) 56

Step-by-step explanation:

Answer:

1018.4 feet

Step-by-step explanation:

The maximum height reached by the rocket can be found by using the formula for the vertex of a parabola.

The vertex of the parabola y = ax^2 + bx + c is given by (-b/2a, c - b^2/4a).

In this case, the equation is y = -16x^2 + 254x + 79.

The maximum height is reached at x = -b/2a = -254/(2*-16) = 7.9375 seconds.

Plugging this value into the equation gives y = -16(7.9375)^2 + 254(7.9375) + 79 = 1018.4 feet.

Therefore, the maximum height reached by the rocket is 1018.4 feet to the nearest tenth of a foot.

Hope this helps!