Find the exact value of tan A in simplest radical form.

Answer:

35

Step-by-step explanation:

Hi,

A simple way to look at this is making the pattern:

Red, Yellow

Red, Yellow, Blue

Red, Yellow, Blue, Green

Red, Yellow, Blue, Green, Pink

Red, Yellow, Blue, Green, Pink, Orange

Red, Yellow, Blue, Green, Pink, Orange, Purple

Red, Yellow, Blue, Green, Pink, Orange, Purple, White

2 + 3 + 4 + 5 + 6 + 7 + 8 = 35

You can otherwise use the Carl Gauss's formula for calculating the sum of increasing consecutive number:

[tex]Sum = \frac{n}{2} \ (a + b)[/tex]

where: n is the total number of rows;

a is the starting number of values and b is the ending number.

Hence, in this case:

n = 7

a = 2

b = 8

[tex]Total\ number\ of\ candles\ = \frac{7}{2}\ (2+8)\\ =35[/tex]

Final answer:

By analyzing the color pattern, which adds one new color each step and concludes with white being the eighth color, there are 8 total candles on the cake.

Explanation:

To determine how many total candles are on the cake with a color pattern that grows and ends with a white candle, we must first recognize the sequence of colors added as the pattern progresses. The given pattern is red, yellow; red, yellow, blue; red, yellow, blue, green, and continues adding in the order of pink, orange, purple, and finally white.

By this sequence, we can see that the color white will be the last in a set of eight candles (red, yellow, blue, green, pink, orange, purple, white). Since we are looking for the point at which the last candle is white, this indicates that the full sequence has been completed one time entirely. Therefore, there are 8 total candles on the cake.

Answer:

The answer to your question is below

Step-by-step explanation:

a) Yes, Joy can conclude that ΔABC is similar to ΔEDC because

∠ACB ≅ ∠ECD    they are vertical angles and,

∠ABC ≅ ∠EDC    they are right angles

We conclude that ΔABC is similar to ΔEDC  because of the AA postulate.

b)       [tex]\frac{AB}{BC} = \frac{DE}{CD}[/tex]

Solve for AB

        AB = (BC)(DE) / (CD)

Substitution

        AB = 105 (90) / 85

Simplification

        AB = 105(1.06)

Result

        AB = 111.2 ft

Answer:

  d.  about 50 times larger

Step-by-step explanation:

The given expressions for magnitude (M) can be solved for the intensity (I). Then the ratio of intensities is ...

  [tex]\dfrac{I_2}{I_1}=\dfrac{I_0\cdot 10^{M_2}}{I_0\cdot 10^{M_1}}=10^{M_2-M_1}=10^{4.2-2.5}\\\\=10^{1.7}\approx 50[/tex]

The larger earthquake had about 50 times the intensity of the smaller one.

Answer:

Step-by-step explanation:

Assuming the rate of increase in the cost of tuition fee per year is linear. We would apply the formula for determining the nth term of an arithmetic sequence which is expressed as

Tn = a + (n - 1)d

Where

a represents the first term of the sequence.

d represents the common difference.

n represents the number of terms in the sequence.

From the information given,

a = $20500(amount in 2000)

From 2000 to 2018, the number of terms is 19, hence,

n = 19

T19 = 454120

Therefore,

454120 = 20500 + (19 - 1)d

454120 - 20500 = 18d

18d = 433620

d = 433620/18

d = 24090

Therefore, the equation that can be used to find the tuition y for x years after 2000 is expressed as

y = 20500 + 24090(x - 1)

To to estimate the tuition at this college in 2020, the number of terms between 2000 and 2020 is 21, hence

x = 21

y = 20500 + 24090(21 - 1)

y = 20500 + 481800

y = $502300

Using linear interpolation, the yearly increase of tuition was calculated and an equation was formed. Substituting the year 2020 into the equation gave an estimated tuition of $47,855.60.

To write an equation that represents the tuition cost y for x years after 2000, we use two given data points: in 2000, tuition was $20,500 (which means when x=0, y=$20,500), and in 2018, tuition was $45,120 (when x=18, y=$45,120). To find the rate of change, we calculate the slope by finding the difference in tuition and dividing it by the difference in years:

Slope (m) = (Y2 - Y1) / (X2 - X1) = ($45,120 - $20,500) / (18 - 0) = $24,620 / 18 ≈ $1,367.78

Now we can write the equation of the line in slope-intercept form (y = mx + b), where b is the initial tuition in the year 2000:

Equation: y = 1367.78x + 20,500

To estimate the tuition in 2020, set x to 20:

y = 1367.78(20) + 20,500 = $27,355.60 + 20,500 = $47,855.60

Therefore, the estimated tuition in 2020 is $47,855.60.

Answer:

x = 3

JK = 13

KL = 13

JL = 26

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Algebra I

Step-by-step explanation:

Step 1: Define

K is midpoint JL. Use midpoint definition.

JK = 2x + 7

KL = 4x + 1

JK = KL

2x + 7 = 4x + 1

Step 2: Solve for x

Step 3: Find

JK

KL

JL

The variable x is found to equal 3. Substituting x=3 into the expressions for JK and KL, both are found to equal 13. The entire line segment JL is then found to equal 26.

In this mathematics problem, we are given that K is the midpoint of JL. This implies that the segments JK and KL are of equal length, thus JK=KL.

So, we can set the two given expressions equal to each other: 2x+7=4x+1. Solving this equation for x, we get that x=3.

Substituting x=3 into the expressions for JK and KL: JK=2x+7=2(3)+7=13, and KL=4x+1=4(3)+1=13.

To find JL, we simply add JK and KL together, since JKL is a line segment with K being the midpoint. Hence, JL=JK+KL=13+13=26.

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Answer: the second one

Step-by-step explanation:

Answer:

40/13 hours

Step-by-step explanation:

Let 1 represent the room being painted.

1/5 would represent 1/5 of a room painted in 1 hour.

1/8 would represent 1/8 of a room being painted in 1 hour

But they are are working together so

(1/5)x + (1/8)x = 1

The lowest common denominator is 5 * 8 = 40

[(1/5)x * 40 + (1/8)x * 40] = 1

8x + 5x = 40

13x = 40

x = 40/13

x = 3.077 hours

But you want a fraction as your answer so it is 3 1/13 or 40/13

You should notice that 3.077 is smaller than the lowest amount of time both of them use. That always happens with these questions.

Final answer:

Rachel and Barbara would take ⅜shy hours to paint the living room together, as they have a combined work rate of 0.325 room/hour calculated from their rates of 1 room per 5 hours and 1 room per 8 hours, respectively.

Explanation:

To find out how many hours it would take Rachel and Barbara to paint the living room together, we can use the concept of work rate. Rachel's work rate is ⅑or (1 room per 5 hours), and Barbara's work rate is ⅑or (1 room per 8 hours). To find their combined work rate, we add their rates together.

Rachel's rate: ⅑or (1 room/5 hours) = 0.2 room/hour
Barbara's rate: ⅑or (1 room/8 hours) = 0.125 room/hour
Combined rate: 0.2 + 0.125 = 0.325 room/hour

To find out how long it takes them to paint the room together, we can set up the equation: 1 room = (0.325 room/hour) × (hours). Solving for 'hours' gives us:

Hours = ⅑or / 0.325
Hours = ⅑or / (⅜shy / 1)
Hours = ⅜shy × 1
Hours = ⅜shy
Therefore, it would take them ⅜shy hours to complete the painting together.

Answer:

x = 110°

Step-by-step explanation:

I think that CHORDS AC & BD are intersecting at centre of the circle. If it is so then,

Since, measure of central angle of a circle is equal to the measure of its corresponding arc or vice versa.

Answer:

The answer to your question is x = 2.5

Step-by-step explanation:

We know that lines r and s are parallel so angles 1 and 2 are corresponding angles. Corresponding angles measure the same.

                                      m∠1 = m∠2

Substitution

                                40 - 4x = 50 - 8x

Solve for x

                               8x - 4x = 50 - 40

                                       4x = 10

                                         x = 10/4

                                         x = 2.5

The length of MG is 56 for the given similar triangles.

Step-by-step explanation:

Let us consider EML and EGF is two similar triangles.

From the triangle,

EG=5x+2.

EM=16.

EL=28.

EF=126.

According to similar triangle property,

triangle ratio= [tex]\frac{EM}{EG} =\frac{EL}{EF} =\frac{ML}{GF} .[/tex]

[tex]\frac{16}{5x+2}=\frac{28}{126}[/tex].

126(16)=28(5x+2).

2016=140x+56.

140x=1960.

x=14.

⇒EG=5(14)+2.

EG=70+2.

EG=72.

To find the length of MG,

EG=EM+MG.

MG=EG-EM.

=72-16.

MG=56.

The question, asking for the length of a segment within a triangle, belongs to the field of Mathematics and is most likely at a High School level. However, based on the given information, an exact answer cannot be provided.

Unfortunately, the provided information is not sufficient enough to find the length of MG. The relationship EGF ~ EML indicates that the triangles EGF and EML are similar, meaning their corresponding sides are proportional. To find the length of MG (a segment within triangle EML), we would typically use a known length from triangle EGF and the scale factor between the two triangles. However, without these additional specifics, we cannot provide an exact value for the length of MG.

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Answer:

64.08

Step-by-step explanation:

To find the angle of elevation of the sun, we use the tangent function with the height of the flagpole and the length of the shadow. Calculating the arctangent of the ratio of the height to the shadow length and converting it to degrees gives us the angle of elevation.

The question asks about the angle of elevation of the sun when casting the shortest shadow of a flagpole. We can use trigonometric functions to find this angle using the height of the flagpole and the length of the shadow. The height of the flagpole is 282 ft and the length of the shadow is 137 ft.

To find the angle of elevation (θ), we use the tangent function, which is the ratio of the opposite side (height of the flagpole) to the adjacent side (length of the shadow). So, tangent(θ) = opposite/adjacent = 282 ft / 137 ft.

Now, we will calculate the angle using the inverse of the tangent function, also known as arctangent.

θ = arctan(282/137)

To find the angle in degrees, we use a calculator to compute arctan(282/137). After computing, we round the result to the nearest whole number to find the angle of elevation to the nearest degree.

Two pounds of gummy bears will cost $2.67. The unit rate for each pounds of gummy bears is measured in dollars.

Word problems in mathematics involve the use of mathematical concepts and arithmetic operations to solve real-life cases. It involves a careful understanding of the problem you want to solve.

From the parameters given:

By cross multiplying, we have:

[tex]\mathbf{x = \dfrac{2 \ pounds \times \$4.00}{\$3.00}}[/tex]

x = $2.67

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Final answer:

The cost for two pounds of gummy bunnies is $2.66, with the unit rate being $1.33 per pound after rounding to two decimal places.

Explanation:

To determine how much two pounds of gummy bunnies will cost at Joan's candy emporium, we first need to calculate the unit rate of the gummy bunnies that are selling for $4.00 per three pounds. The unit rate is found by dividing the total cost by the number of pounds:

Unit Rate = Total Cost / Number of Pounds

Unit Rate = $4.00 / 3 pounds = $1.33 per pound (rounded to two decimal places)

Now that we have the unit rate, we can determine the cost for two pounds of gummy bunnies:

Cost for Two Pounds = Unit Rate x Number of Pounds

Cost for Two Pounds = $1.33 per pound x 2 pounds = $2.66 (rounded to two decimal places)

The differential equation provided is transformed to M(x,y) dx + N(x,y) dy = 0 with M(x,y) = [tex]-4x^4 - 3y[/tex]and N(x,y) = 3x + [tex]2y^4[/tex], setting the stage for verifying its exactness through partial differentiation.

To address the student's query about finding a function F(x,y) whose differential fits the given differential equation, we'll first transform the differential equation dy/dx =[tex](-4x4-3y)/(3x+2y4)[/tex] into the form M(x,y) dx + N(x,y) dy = 0.

This transformation requires us to regard the equation as a differential one, implying:

This representation simplifies the process of checking for exactness, which necessitates partial differentiation and comparison of ∂M/∂y and ∂N/∂x.

Should these partial derivatives be equal, the differential is exact, enabling the determination of the function F(x, y) through integration.

Answer:

Andre can answer 1.5 multiplication facts in each second.

Step-by-step explanation:

Given:

Number of multiplication facts completed = 135

Number of second took to complete multiplication facts = 90 seconds.

We need to find the Number of facts answered per second.

Solution;

Now we know that;

In 90 seconds = 135 multiplication facts.

So in 1 second = Number of fact answered in 1 second

By Using Unitary method we get;

Number of fact answered in 1 second = [tex]\frac{135}{90}=1.5[/tex]

Hence Andre can answer 1.5 multiplication facts in each second.

Answer: the plant is 6 1/4 inches above the​ ground.

Step-by-step explanation:

In her garden Pam plants the seed 5 and one fourth inches below the ground. Converting 5 and one fourth inches to improper fraction, it becomes 21/4 inches.

After one month the tomato plant has grown a total of 11 and one half inches. Converting 11 and one half inches to improper fraction, it becomes 23/2 inches.

The height of the the plant above the​ ground would be

23/2 - 21/4 = (46 - 21)/4 = 25/4

Converting to mixed fraction, it becomes

6 1/4 inches

Answer:

1) The possible outcomes 2) Quantitative 3) The explanation to those outcome 4) Qualitative

Step-by-step explanation:

1) The response variable is a measurable variable, i.e. also called the dependent variable. In the study, they will represent the possible outcome.

E.g.

Suppose the study "Practicing enhances technique", the amount of hours will be the response variable.

2) Is the response variable qualitative or​ quantitative? Since it is measured, it's a quantitative.

In our example, our response variable would be hours, how many hours is necessary to display some enhancement?

3) What is the explanatory​ variable?

The explanatory ones, or also independent variables offer explanations to the results the response variables have shown.

In our example, the level of training (low, mid, hard) would be the explanatory one.

4) Is the explanatory variable qualitative or​ quantitative?

In our example, the explanatory response or independent one is qualitative since to classify the training as low, middle or harder is to classify them as categorical then it's qualitative.

Answer:

The concepts being studied

Step-by-step explanation:

What are variables? ap e x

Answer:

(a) 0.0024

(b) 0.0012

(c) 0.0006

(d) 0.0004

Step-by-step explanation:

The total number of possible integers when any number is selected is 10 (i.e from 0 - 9). When four number integers are selected, the total number of sample sample will be;

                                   10 × 10 × 10 × 10 = 10,000

The sample space = 10,000

To know the possible ways of selecting the given four digits, we will use permutation.

                                 [tex]^{n}P_{r} = \frac{n!}{(n-r)!}[/tex]

To get the probability,

[tex]Probability \ of \ winning (Selected \ numbers) = \frac{number\ of\ possible\ outcomes\ of\ selected\ numbers}{sample\ space}[/tex]

(a) When 6,7,8,9 are selected, n = 4 , r = 4

The possible ways of selecting 6,7,8,9 is;

                                    [tex]^{4}P_{4} = \frac{4!}{(4-4)!}[/tex]

                                            [tex]= \frac{4!}{(0)!}[/tex]

but 0! = 1

                                     [tex]^{4}P_{4} = 4![/tex]

                                     = 4 × 3 × 2 × 1 = 24

                 [tex]Prob (6,7,8,9) = \frac{24}{10000} = 0.0024[/tex]

(b) When 6, 7, 8, 8 are selected,

The possible ways of selecting 6,7,8,8 is;

                                     [tex]= \frac{4!}{1! \ 1! \ 2!}[/tex]

                                     [tex]= \frac{4!}{2!}[/tex]  

                                     [tex]=\frac{4 * 3 * 2 * 1}{2 * 1}[/tex]

                                     = 12

             [tex]Prob (6,7,8,8) = \frac{12}{10000} = 0.0012[/tex]

(c) When 7, 7, 8, 8 are selected,

The possible ways of selecting 7,7,8,8 is;

                                     [tex]= \frac{4!}{2! \ 2!}[/tex]

                                     [tex]=\frac{4 * 3 * 2 * 1}{(2 * 1)(2 * 1)}[/tex]

                                     = 6

             [tex]Prob (7,7,8,8) = \frac{6}{10000} = 0.0006[/tex]

(d) When 7, 8, 8, 8 are selected,

The possible ways of selecting 7,8,8,8 is;

                                    [tex]= \frac{4!}{1! \ 3!}[/tex]    

                                    [tex]=\frac{4 * 3 * 2 * 1}{3 * 2 * 1}[/tex]    

                                    = 4

             [tex]Prob (7,8,8,8) = \frac{4}{10000} = 0.0004[/tex]

Answer:

C

Step-by-step explanation:

35miles = 10days

x = 1day

10x = 35*1

x = 35/10

x = 3.5miles per day

Answer:

Probability p( selecting 8 cities alphabetically) = 2.48×10^-5

Step-by-step explanation:

Number of possible ways of choosing 8 cities=Permutation P(n,k) = n!/(n-k)!

P(8,8)= 8!/(8-8)! = 8! = 40,320

Probability (selecting 8 cities alphabetically) = 1/40320 = 2.48×10^-5

Answer:

The answer is 0.00002480

Step-by-step explanation:

From the question stated let us recall the following statement.

The number of cities the tourist wants to visit is =8

Now,

If the route is selected randomly, what is the probability that the cities she visits are in alphabetical order.

Therefore,

The probability that she visits the cities in alphabetical order = 1/8!

There are 8! = 40320 ways in which these cities can be visited

P(Visiting in alphabetical order) = 1/40320 = 0.00002480

Answer:

[tex]k=250[/tex]

Step-by-step explanation:

If y varies inversely with x, then

[tex]y=\dfrac{k}{x},[/tex]

where k is the constant of variation or the constant of inverse proportionality.

Since [tex]x=2.5[/tex] when [tex]y=100,[/tex] you have that

[tex]100=\dfrac{k}{2.5}[/tex]

Evaluate k:

[tex]k=100\cdot 2.5\\ \\k=250[/tex]

and

[tex]y=\dfrac{250}{x}[/tex]

Answer:

8/9 of a cup

Step-by-step explanation:

8 cups of icecream to be divided among 9 people, so they can't have full cups.

Each person will get 8/9 of a cup

Answer:

  7.5

Step-by-step explanation:

For function f(x), the average rate of change between x=a and x=b is given by ...

  average rate of change = (f(b) -f(a))/(b -a)

For your function, this will be ...

  average rate of change = ((2^5 +3) -(2^1 +3))/(5 -1) = (35 -5)/4

  average rate of change = 7.5

Answer: [tex]x=5[/tex]

Step-by-step explanation:

The area of a rectangle can be found with the following formula:

[tex]A=lw[/tex]

Where "l" is the length and "w" is the width.

In this case you can identify in the figure given in the exercise that:

[tex]l=4x-2\\\\w=3[/tex]

You know that the area of that rectangle is the following:

[tex]A=54[/tex]

Therefore, knowing those values, you can substitute them into the formula and then you must solve for "x" in order to find its value. You get that this is:

[tex]54=(4x-2)(3)\\\\54=12x-6\\\\54+6=12x\\\\\frac{60}{12}=x\\\\x=5[/tex]

Answer:

200

Step-by-step explanation:

50 doubles to 100 and then doubles again to 200.

Answer:

a) The demand function is

[tex]q(p) = -4 p + 107[/tex]

b) The nightly revenue is

[tex] R(p) = -4 p^2 + 107 p [/tex]

c) The profit function is

[tex]P(p) = -4 p^2 + 133.75 p - 939 [/tex]

d) The entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.

Step-by-step explanation:

a) Lets find the slope s of the demand:

[tex] s = \frac{79-43}{7-16} = \frac{36}{-9} = -4 [/tex]

Since the demand takes the value 79 in 7, then

[tex]q(p) = -4 (p-7) + 79 = -4 p + 107[/tex]

b) The nightly revenue can be found by multiplying q by p

[tex]R(p) = p*q(p) = p*( -4 p + 107) = -4 p^2 + 107 p[/tex]

c) The profit function is obtained from substracting the const function C(p) from the revenue function R(p)

[tex]P(p) = R(p) - C(p) = p*q(p) = -4 p^2 + 107 p - (-26.75p + 939) = \\\\-4 p^2 + 133.75 p - 939[/tex]

d) Lets find out the zeros and positive interval of P. Since P is a quadratic function with negative main coefficient, then it should have a maximum at the vertex, and between the roots (if any), the function should be positive. Therefore, we just need to find the zeros of P

[tex]r_1, r_2 = \frac{-133.75 \,^+_-\, \sqrt{133.75^2-4*(-4)*(-939)} }{-8} = \frac{-133.75 \,^+_-\, 53.526}{-8} \\r_1 = 10.03\\r_2 = 23.41[/tex]

Therefore, the entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.

We found the linear demand equation as q(p) = -4p + 107. The nightly revenue R(p) is R(p) = -4p^2 + 107p, and the profit P(p) is P(p) = -4p^2 + 133.75p - 939. Setting the profit equal to zero, we found the club breaks even at entrance fees $5.32 and $44.31.

In this question, we're asked to formulate linear demand, revenue, and profit equations, and find the entrance fees for break even. First, let's find the linear demand equation.

To obtain this demand equation, we can use the two given points ($7, 79) and ($16, 43) and find the slope (rate of change) equals (43-79) / (16-7) = -4. Hence the demand equation will be q(p) = -4p + b. To find b, substitute one of the points into the equation, for example ($7, 79), we get b=107, hence q(p) = -4p + 107.

Next, let's find the nightly revenue R (p), which is simply the product of the entrance fee and the number of guests, hence R(p) = p * q(p) = p (-4p + 107) = -4p^2 + 107p.

 The profit P(p) is the difference between revenue and costs, hence P(p) = R(p) - C(p) = -4p^2 + 107p - (-26.75p + 939) = -4p^2 + 133.75p - 939.

Finally, for Swing Haven to break even, the profit must be zero. By setting P(p) = 0 and solving the equation -4p^2 + 133.75p - 939 = 0, we get the two solutions p = 5.3216 and p = 44.3031, or roughly $5.32 and $44.31.

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Answer:A randomization

Step-by-step explanation:Randomization in scientific experiments is a sampling method in which the participants or researchers are chosen randomly and assigned a treatment.

Here the participants or researchers do not know for sure which treatment is better.

Randomization reduces any possible bias responses to a minimal.

Answer:

Fred used 8 roses to make the bouquet.

Step-by-step explanation:

Let 'x' roses be used to make a bouquet.

Given:

Cost of carnations = $5.25

Cost of 1 rose = $1.68

Total cost of the bouquet = $18.69

Cost of 1 rose = $1.68

Therefore, using unitary method, the cost of 'x' roses is given as:

Cost of 'x' roses = [tex]1.68x[/tex]

Now, as per question:

Cost of carnations + Cost of 'x' roses = Total cost of the bouquet

[tex]5.25+1.68x=18.69\\\\1.68x=18.69-5.25\\\\1.68x=13.44\\\\x=\frac{13.44}{1.68}\\\\x=8[/tex]

Therefore, Fred used 8 roses to make the bouquet.

Fred used 8 roses in the bouquet.

To solve the problem, we need to find out how many roses Fred used, given the total cost of the bouquet and the cost of the carnations and each rose.

Let's denote the number of roses as ( r ).

The cost of one rose is $1.68. Therefore, the cost of ( r ) roses is ( 1.68r ).

The cost of the carnations is given as $5.25.

The total cost of the bouquet is $18.69.

We can set up the equation to represent the total cost of the bouquet as the sum of the cost of the carnations and the cost of the roses:

[tex]\[ 5.25 + 1.68r = 18.69 \][/tex]

Now, we need to solve for [tex]\( r \):[/tex]

Subtract $5.25 from both sides of the equation to isolate the term with [tex]\( r \):[/tex]

[tex]\[ 1.68r = 18.69 - 5.25 \][/tex]

[tex]\[ 1.68r = 13.44 \][/tex]

Next, divide both sides by $1.68 to solve for r :

[tex]\[ r = \frac{13.44}{1.68} \][/tex]

[tex]\[ r = 8 \][/tex]

Part 1: The function that models the height of the basketball is [tex]\( h(t) = -16t^2 + 15t + 6.5 \).[/tex]

Part 2: The basketball hits the ground after approximately 0.97 seconds.

Part 3: The basketball reaches its maximum height at approximately 0.47 seconds.

Part 4: The maximum height of the basketball is approximately 11.39 feet.

Part 1: To model the height of the basketball, we use the given function [tex]\( h(t) = -16t^2 + v0t + h0 \)[/tex] with the initial velocity [tex]\( v0 = 15 \)[/tex] ft/sec and the initial height [tex]\( h0 = 6.5 \)[/tex] feet. Plugging in these values, we get the function:

[tex]\[ h(t) = -16t^2 + 15t + 6.5 \][/tex]

Part 2: To find the time it takes for the basketball to hit the ground, we set the height function equal to zero and solve for [tex]\( t \)[/tex]:

[tex]\[ -16t^2 + 15t + 6.5 = 0 \][/tex]

Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),[/tex]where [tex]\( a = -16 \), \( b = 15 \)[/tex], and [tex]\( c = 6.5 \)[/tex], we get two solutions. We discard the negative solution because time cannot be negative, and we round the positive solution to the nearest hundredth:

[tex]\[ t = \frac{-15 \pm \sqrt{15^2 - 4(-16)(6.5)}}{2(-16)} \] \[ t = \frac{-15 \pm \sqrt{225 + 416}}{-32} \] \[ t = \frac{-15 \pm \sqrt{641}}{-32} \] \[ t \approx \frac{-15 + 25.31}{-32} \] \[ t \approx 0.97 \text{ seconds} \][/tex]

Part 3: To find when the basketball reaches its maximum height, we need to find the vertex of the parabola. The time coordinate of the vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] is given by [tex]\( t = -\frac{b}{2a} \)[/tex]. For our function, [tex]\( a = -16 \)[/tex]and [tex]\( b = 15 \)[/tex], so:

[tex]\[ t = -\frac{15}{2(-16)} \] \[ t = \frac{15}{32} \] \[ t \approx 0.47 \text{ seconds} \][/tex]

Part 4: To find the maximum height, we substitute the time at which the maximum height is reached back into the height function:

[tex]\[ h(0.47) = -16(0.47)^2 + 15(0.47) + 6.5 \] \[ h(0.47) \approx -16(0.2209) + 7.05 + 6.5 \] \[ h(0.47) \approx -3.5344 + 7.05 + 6.5 \] \[ h(0.47) \approx 11.39 \text{ feet} \][/tex]

Therefore, the basketball reaches a maximum height of approximately 11.39 feet after approximately 0.47 seconds.

Answer:

The answer to your question is dAC = 10 units

Step-by-step explanation:

Data

From the graph, take the coordinates of the points A and C.

A (3, - 1)

C (-5, 5)

Process

Use the formula of the distance between two points to find the length

d = [tex]\sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}[/tex]

x1 = 3    y1 = -1

x2 = -5  y2 = 5

Substitution

dAC = [tex]\sqrt{(-5 - 3)^{2} + (5 + 1)^{2}}[/tex]

Simplification

dAC = [tex]\sqrt{(-8)^{2} + (6)^{2}}[/tex]

dAC = [tex]\sqrt{64 + 36}[/tex]

dAC = [tex]\sqrt{100}[/tex]

dAC = 10 units