You throw a ball straight up from a rooftop 70 feet high with an initial speed of 80

feet per second. The functions s(t)=â16t2+80t+70models theâ ball's height above theâ ground,s(t)

inâ feet, t seconds after it was thrown. During which time period will theâ ball's height exceed that of theâ rooftop?

When will the ball exceed the height of theâ rooftop?

Answer:

(-6,0)

Step-by-step explanation:

An equation of a line can be modeled as y = mx + b where m is slope and b is y-intercept.

For the line r, we can model the equation as y = mx + 3 since the line intersects y-axis at (0,3) as seen in the attachment.

For the line t, we can model the equation as y = mx - 6 as the problem gives y-intercept for line t equal to -6. Hence, the line t intersects y-axis at (0,6)

Next, we have to find the slope of line t by finding the slope of line r in the attachment. Apply the rise over run by counting the steps, you can see in the attachment that I put to learn how to count rise and run of a line. Also note that the value in attachment here is a scalar quantity, meaning only magnitude, no direction.

So we will have the slope of -1 since a line graph is heading down so the output decreases as input increases. Therefore, we know that m = -1 for both lines. Therefore, for the line t, we can model the new equation to:

[tex]\displaystyle{y=-x-6}[/tex]

Then we find the x-intercept of the line by letting y = 0. Thus,

[tex]\displaystyle{0=-x-6}\\\\\displaystyle{x=-6}[/tex]

Therefore, the x-intercept of line t is at (-6,0).

Answer:

(-6,0)

Step-by-step explanation:

We need to determine the equation of line r first to find the x-intercept of line t.

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

[tex]slope = \frac{y_2 - y_1}{x_2 -x_1}[/tex]

For line r passing through (3, 0) and (0, 3), the slope is:

[tex]slope = \frac{3 - 0}{0 - 3} = -1[/tex]

Since line t has the same slope as line r, its slope is also -1.

The equation of a line in slope-intercept form (y = mx + b) is determined by its slope (m) and y-intercept (b).

We know that the slope (m) of line t is -1, and the y-intercept (b) is -6. Substituting these values into the slope-intercept form, we get:

y = -x - 6

To find the x-intercept, we set y = 0 and solve for x:

0 = -x - 6

Adding x to both sides:

x = -6

Therefore, the x-intercept of line t is (-6,0).

The lengths of the sides and angles of the triangles, obtained using Pythagorean Theorem and trigonometric ratios are;

D. 19.2

C. 5.3

F. 2·√5

G. √3

H. 8·√3

K. 18.9

O. 15.5

N. 14.5

M. 41.4°

I. 74.1°

The Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the legs of the triangle.

D. x = √(15² + 12²) ≈ 19.2

C. x = √(10² - 8.5²) ≈ 5.3

F. x = 2·√(5) (Special triangles)

G. tan (60°) = 30/x = √3

x = 30/√3 = 10·√3

H. x/16 = (√3)/2

x = 16 × (√3)/2 = 8·√3

x = 8·√3

K. tan(34°) = x/28

x = 28 × tan(34°) ≈ 18.9

O. cos(55°) = x/27

x = 27 × cos(55) ≈ 15.5

N. sin(16°) = 4/x

x = 4/(sin(16°) ≈ 14.5

M. x = arccos(9/12) ≈ 41.4°

I. x = arcsin(25/26) ≈ 74.1°

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The simplification of the expression, ∛8x⁴y⁸ / 125 is [tex]\frac{2\sqrt[3]{x^{4}y^{8} } }{5}[/tex].

The exponential expression can be solved as follows:

Therefore, let's simplify the expression.

To simplify the expression we have to deal with the exponential by applying it laws and also the cube root.

∛8x⁴y⁸ / 125

let's solve them individually,

125 = 5³ = 5 × 5 × 5

8 = 2³ = 2 × 2 × 2

Therefore,

∛8x⁴y⁸ / 125 = [tex]\frac{2\sqrt[3]{x^{4}y^{8} } }{5}[/tex]

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Answer:

Center: (-3, 2)

Radius: 4

Step-by-step explanation:

x2 + 6x + y2 - 4y = 3​

x² + 6x + 9 + y² - 4y + 4 = 3 + 9 + 4

(x + 3)² + (y - 2)² = 16

(x + 3)² + (y - 2)² = 4²

Center: (-3, 2)

Radius: 4

The measure in radians for the central angle is approximately 0.7 radians.

To find the measure in radians for the central angle of a circle, we can use the formula:

θ = s / r

where θ is the central angle in radians, s is the intercepted arc length, and r is the radius of the circle.

In this case, the radius is given as 8 cm and the intercepted arc length is 5.6 cm.

Plugging these values into the formula, we have:

θ = 5.6 cm / 8 cm

Simplifying the expression:

θ = 0.7

We may use the following formula to get the centre angle of a circle's measurement in radians:

= s / r

s is the length of the intercepted arc, r is the circle's radius and is the centre angle in radians.

The intercepted arc length in this instance is 5.6 cm, while the radius is specified as 8 cm.

When these values are plugged into the formula, we get:

θ = 5.6 cm / 8 cm

Condensing the phrase:

θ = 0.7

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Answer: 360 containers

Step-by-step explanation: 1 5/8 in decimal form is 1.625 pounds. Divide 585 by 1.625 pounds you get 360. It takes 360 containers to hold 585 pounds of fruit

Following a translation of (x, y) (x - 1, y + 2) and a dilation by a scale factor of 2 centred at the origin, the coordinates of T are T''(8, 8).

Following translation and dilation, we must take the following actions to determine the new coordinates of point T:

1) Translation: (x, y) = (x - 1, y + 2).

T' = (5 - 1, 2 + 2)

= (4, 4) applies the translation to the original coordinates of T.

2) Dilation: Scale factor of 2 centered at the origin

Multiply the translated coordinates by the scale factor of 2:

T'' = (2 × 4, 2 × 4)

= (8, 8)

Therefore, the coordinates of T after the translation (x, y) → (x - 1, y + 2) followed by a dilation by a scale factor of 2 centered at the origin are T''(8, 8).

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The sum of the expressions 12√23x¹³ + 14√23x¹³ is in its simplest form if x > 0 is 26x⁶√23x

From the question, we have the following parameters that can be used in our computation:

12\sqrt{23x^{13}}+14\sqrt{23x^{13}}

Express the summation expression, properly

So, we have

12√23x¹³ + 14√23x¹³

Next, we add the terms of teh expression

Using the above as a guide, we have the following:

12√23x¹³ + 14√23x¹³ = 26√23x¹³

Take the square root of x¹³

12√23x¹³ + 14√23x¹³ = 26x⁶√23x

The above expression is in its simplest form if x > 0.

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Answer:

  y = -10·sin(πt/5) +12

Step-by-step explanation:

You want the equation of the height of a rider of a Ferris wheel that has a radius of 10 and is 2 off the ground, with a period of 10 seconds, moving downward, starting from even with the center.

The general form of the equation will be ...

  y = A·sin(2πt/T) + B

where A is a scale factor that is based on the radius and initial direction, and B is the height of the center of the wheel above the ground.

We assume that 2 [units] off the ground means the low point of the travel is at that height. Then the middle of the wheel is those 2 [units] plus the radius of the wheel:

  B = 2 + 10 = 12

The scale factor A will be the radius of the wheel, made negative because the initial direction is downward from the initial height. That is, ...

  A = 10

The period (T) is given as 10 seconds.

Putting these parameters together gives ...

  y = -10·sin(2πt/10) +12

  y = -10·sin(πt/5) +12

__

Additional comment

We wonder if this wheel is really only 20 inches (20 in) in diameter, as that dimension seems suitable only for a model. We suspect it is probably 20 meters (20 m) in diameter.

Sometimes "m" is confused with "in" when it is written in Roman font and reproduced with poor resolution.

<95141404393>

The expression that represents a purely imaginary number is:

[tex]3i^5 + i (2 - 4i^3) - 2 (-3 + i)[/tex]

Option D is the correct answer.

We have,

A pure imaginary number is a number that does not have a real part and can be written in the form of bi, where b is a real number and i is the imaginary unit (√(-1)).

We have,

[tex]3i^5 + i (2 - 4i^3) - 2 (-3 + i)[/tex]

In this expression, we can simplify and rewrite it as:

[tex]3i^5 + i(2 - 4i^3) + 6 - 2i[/tex]

Now, let's evaluate the powers of i:

[tex]i^5 = i^{4 + 1} = i^4 \times i = 1 \times i = i\\i^3 = i^{2 + 1} = i^2 \times i = (-1) \times i = -i[/tex]

Substituting these values back into the expression:

[tex]3i^5 + i(2 - 4i^3) + 6 - 2i = 3i + i(2 - 4(-i)) + 6 - 2i[/tex]

= 3i + i(2 + 4i) + 6 - 2i

= 3i + 2i + 4i² + 6 - 2i

= 3i + 2i + 4(-1) + 6 - 2i

= 5i - 4 + 6 - 2i

= 5i - 2i - 4 + 6

= 3i + 2

The expression 3i + 2 represents a pure imaginary number, as it has no real part (the constant term is 2 and does not involve the imaginary unit i).

Thus,

The expression that represents a purely imaginary number is:

[tex]3i^5 + i (2 - 4i^3) - 2 (-3 + i)[/tex]

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The product of given two binomials which are (x-7)(x-7) is equal to x² - 14x + 49.

To multiply the binomials (x-7)(x-7), we use the distributive property of multiplication, which states that:

(a + b)(c + d) = ac + ad + bc + bd

Using this property, we can expand the expression as follows:

(x-7)(x-7) = x(x-7) - 7(x-7)

= x² - 7x - 7x + 49

= x² - 14x + 49

The distributive property of multiplication is a mathematical rule that states that when a number is multiplied by the sum of two or more numbers, it is equivalent to multiplying the number by each of the addends and then adding the products.

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Answer:0

Step-by-step explanation:

so if x = -3 that means u take -3 * 2 and u get -6 because a negative * A positive is negative so -6 + 6 is 0

Answer:

0

Step-by-step explanation:

Just substitute x=-3

2x+6 =

2(-3) + 6 =

-6 + 6 = 0

The results of the product of 12 and 4 decreased by 15 is 33

Product refers to another term which means multiplication of numbers.

product of 12 and 4

= (12 × 4)

decreased by 15 means 15 is subtracted from the expression

= (12 × 4) - 15

Following the rules of PEMDAS

P = parenthesis

E = exponents

M = multiplication

D = Division

A = addition

S =subtraction

So,

(12 × 4) - 15

= 48 - 15

= 33

Hence, 33 is the product of 12 and 4 decreased by 15

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The value of the product of 12 and 4 decreased by 15 is 33

From the question, we have the following parameters that can be used in our computation:

The product of 12 and 4 decreased by 15

The product of 12 and 4 is expressed as

12 * 4

Evaluate

12 * 4 = 48

When decreased by 15, we subtract 15 from both sides of the equation

using the above as a guide, we have the following:

12 * 4 - 15 = 48 - 15

Evaluate

12 * 4 - 15 = 33

Hence, the product of 12 and 4 decreased by 15 is 33

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5. GH= 3

HI= 3x-9 and x=4

HI= 3(4)-9=

= 12-9

= 3

so, GH = HI

Do the same for GJ and JI.

6. 36=2x

18=x

So, the angles are the same.

4x-11=61

4(18)-11=61

So, the sides are also the same meaning the triangles are congruent.

Answer:

 0.12

Step-by-step explanation:

Given the frequencies of occurrence in the diagram, you want to know the probability of A and B.

The probability of an event is the ratio of the number of times it occurs to the total number of trials.

The diagram shows the event (A and B) occurs 6 times out of a total of 6+6+20+20 = 52 trials.

  P(A and B) = 6/52

  P(A and B) ≈ 0.12

<95141404393>

Answer:

Elle should buy 1 gallon of milk for $3.

Step-by-step explanation:

In one gallon of milk, there are four quarts. So we can multiply the number of gallons by four to convert it to quarts.

(1 gallon)x(4 quarts) = 4 quarts in one gallon

4 quarts is greater than 2 quarts so 1 gallon for $3 is a better deal than 2 quarts for $3.

Answer:

KL = 7, so B is the correct answer.

The expression that will represent the volume of the identical rectangular base pyramid is: 2[One-third (5) (0.25) (2)] cubic units.

To calculate for the volume of a rectangular base pyramid, we use the formula:

volume = 1/3 × area of base rectangle × height

volume of one identical pyramid = (1/3 × 5 × 0.5 × 2)cubic units

volume of the two identical pyramid = 2(1/3 × 5 × 0.5 × 2)cubic units.

Therefore, the expression that will represent the volume of the identical rectangular base pyramid is: 2[One-third (5) (0.25) (2)] cubic units.

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Note that in this case, the missing information which is the shortest time is 27 minutes. this is computed using the knowledge of the range.

The range of a collection of data in statistics is the difference between the highest and smallest values, calculated by subtracting the sample maximum and minimum. It's measured in the same units as the data.

Thus,

Range = Longest time - Shortest time

Since

Mean: 39 minutes

Range: 26 minutes

Longest time: 53 minutes

Shortest time = Longest time - Range

⇒  Shortest time = 53 - 26

= 27 minutes.

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The height h of the cylinder is 9 meters.

A cylinder is simply a 3-dimensional shape having two parallel circular bases joined by a curved surface.

The surface area of a cylinder is expressed as;

Surface Area = 2πrh + 2πr²

Given that; the surface area is 104π m² and the radius as 4 m.

We need to solve for the height (h).

Plug the values into the above formula:

Surface Area = 2πrh + 2πr²

104π = 2πrh + 2πr²

104π = 2π( rh + r² )

Divide through by 2π

52 = rh + r²

52 = ( 4 × h) + ( 4² )

52 = 4h + 16

Subtract 16 from both sides

52 - 16 = 4h + 16 - 16

52 - 16 = 4h

36 = 4h

4h = 36

h = 36/4

h = 9 meters

Therefore, the height is 9 meters.

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a) The area of the entire trapezoidal rear window =  882 sq.in.

b) The fractional part of a complete circle is cleared on the rear window by the 18-inch wiper = 5/12

c) The area of the part of the rear window that is cleared by the wiper = 424.12 sq. in.

d) The percent of the area of the entire rear window is cleared by the wiper =  48.09%

We know that the formula for the area of trapezoid,

A = ((a + b) / 2) × h

Here, a = 36 in., b = 48 in. and height of the trapezoid h=21 in

Using above formula, the area of the entire trapezoidal rear window would be,

A = ((36 + 48) / 2) × 21

A = 882 sq.in.

Here, the 18-inch rear window wiper clears a 150° sector of a circle on the rear window.

We know that the measure of entire circle = 360°

So, the fractional part of a complete circle is cleared on the rear window by the 18-inch wiper would be,

150° / 360° = 5/12

Now we need to find the area of the part of the rear window that is cleared by the wiper.

We know that the formula for the area of sector of a circle is:

A = (θ/360) × πr²

Here, the central angle θ = 150° and radius r = 18 in.

A = (θ/360) × πr²

A = (150/360) × π × 18²

A = 424.12 sq. in.

Now we need to find the percent of the area of the entire rear window is cleared by the wiper.

P = [(area of the part of the rear window cleared by the wiper) / (area of the entire trapezoidal rear window)] × 100

P = (424.12 / 882) × 100

P = 48.09%

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Answer:

last option

Step-by-step explanation:

let g = 3,

9/5 + 7/2(3)

9/5 + 21/2

adjust to a common denominator of ten to get;

18/10 + 105/10

= 123/10 = 12 3/10

so the last option !

The volume of the shape for Jackson's ice cream is given as follows:

11.85 in³.

The volume of a cone of radius r and height h is given by the equation presented as follows:

V = πr²h/3.

Hence the volume of the cone in this problem is given as follows:

V = 3.14 x 1.5² x 5/3

V = 11.78 in³.

The volume of an hemisphere of radius r is given as follows:

V = 2πr³/3.

Hence the volume of sphere is:

V = 2π x 1.5³/3

V = 7.07 in³.

Hence the total volume is given as follows:

11.78 + 7.07 = 11.85 in³.

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1/4π radians is approximately equal to 45 degrees.

To convert a value from radians to degrees, multiply the value by 180/π. For example, to convert π/4 radians to degrees, multiply π/4 by 180/π to get 45 degrees. This conversion is useful for converting angles between the two units of measurement.

Degrees = Radians × 180/π.

Therefore, to convert 1/4π radians to degrees, we have:

Degrees = (1/4π) × (180/π) ≈ 45°

Hence, 1/4π radians is approximately equal to 45 degrees.

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Answer:

x = -5

Step-by-step explanation:

-5 - 15x - 10 = -4x - 8x

-5 - 10 = -4x - 8x + 15x

-15 = -12x + 15x

-15 = 3x

x = -5

The rate is 5.2%

After 10 years, the money will amount to  £4150.

We know that the compound interest can be obtained from;

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

The rate is obtained from;

I = A - P

I = interest

A = amount

P = principal

r = rate

n = Number of times compounded

t = time in years

Then;

I = £2630.00 -  £2500.00

I = £130

I = PRT/100

130 = 2500 * R * 1/100

R = 130 * 100/2500

R = 5.2%

After 10 years;

[tex]A = 2500 ( 1 + 0.052)^{10}[/tex]

A = £4150

Thus this is the amount after 10 years

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Answer:

{0.0853,0.1531}

Step-by-step explanation:

[tex]\displaystyle CI=\hat{p}\pm z\sqrt{\frac{\hat{p}\bigr(1-\hat{p}\bigr)}{n}}[/tex]

[tex]\displaystyle CI_{98\%}=\frac{59}{495}\pm2.326\sqrt{\frac{\frac{59}{495}\bigr(1-\frac{59}{495}\bigr)}{495}}\approx\{0.0853,0.1531\}[/tex]

The distance between each pair of points include the following:

44. 2.8 units.

45. 5.8 units.

46. 12.0 units.

47. 16.6 units.

48. 7.1 units.

49. 4.5 units.

In Mathematics and Geometry, the distance between two (2) end points that are on a coordinate plane can be calculated by using the following mathematical equation:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

x and y represent the data points (coordinates) on a cartesian coordinate.

Question 44.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(3 - 1)² + (2 - 4)²]

Distance = √[(2)² + (-2)²]

Distance = √8

Distance = 2.8 units.

Question 45.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(-2 + 7)² + (0 - 3)²]

Distance = √[(5)² + (-3)²]

Distance = √34

Distance = 5.8 units.

Question 46.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(-6 - 2)² + (-2 - 7)²]

Distance = √[(-8)² + (-9)²]

Distance = √145

Distance = 12.0 units.

Question 47.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(1 - 8)² + (-8 - 7)²]

Distance = √[(-7)² + (-15)²]

Distance = √274

Distance = 16.6 units.

Question 48.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(-5 - 2)² + (-4 + 5)²]

Distance = √[(-7)² + (1)²]

Distance = √50

Distance = 7.1 units.

Question 49.

By substituting the given end points into the distance formula, we have the following;

Distance = √[(5 - 3)² + (5 - 1)²]

Distance = √[(2)² + (4)²]

Distance = √20

Distance = 4.5 units.

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In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

Diagram of how a function relates two relations.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products.

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write

(

0

,

100

]

. We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

The smallest term from the interval is written first.

The largest term in the interval is written second, following a comma.

Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.

Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

Answer:

26 inches

Step-by-step explanation:

The snail starts 20 inches above the ground: +20

The snail slips (goes down) 6 inches: 20 - 6 = 14

The snail goes up 12 inches: 14 + 12 = 28

The snail is 28 inches above the ground.

Hope this helps!