Write a formula that describes the value of an initial investment of $100 that loses its value at a rate of 80% per year compounded 6 times. per year.
​

Answer:

2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.

Step-by-step explanation:

Given model of cost:

[tex]C(x, y) = 7,000 + 100x^2 + 50y^2[/tex]

Government clean-air subsidies amount for sulfur = 500 $/pound

Government clean-air subsidies amount for lead = 100 $/pound

Subsidies amount for x pounds of sulfur = x500 $

Subsidies amount for y pounds of lead= y100 $

Model of subsidy amount :

[tex]S(x,y)=500x+100y[/tex]

Net cost(N) = Cost - Subsidy = C(x,y)-S(x,y)

[tex]N=7,000 + 100x^2 + 50y^2-500x-100y[/tex]..[1]

Differentiating above [1] in with respect to dx :

[tex]\frac{dN}{dx}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dx}[/tex]

[tex]\frac{dN}{dx}=200x-500[/tex]..[2]

Putting [tex]\frac{dN}{dx}=0[/tex]:

[tex]0=200x-500[/tex]

x = 2.5

Now taking second derivative of [2]:

[tex]\frac{d^N}{dx^2}=200[/tex]

[tex]\frac{d^N}{dx^2}>0[/tex] (minima)

Differentiating above [1] in with respect to dy :

[tex]\frac{dN}{dy}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dy}[/tex]

[tex]\frac{dN}{dy}=100y-100[/tex]..[3]

Putting [tex]\frac{dN}{dy}=0[/tex]:

[tex]0=100y-100[/tex]

y = 1

Now taking second derivative of [3]:

[tex]\frac{d^N}{dy^2}=100[/tex]

[tex]\frac{d^N}{dy^2}>0[/tex] (minima)

2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.

The amounts of sulfur and lead the firm should remove are 2.5 pounds and 1 pound respectively to minimize the net cost.

Given to us

cost of controlling emissions, C(x, y) = 7,000 + 100x² + 50y²

amount to $500 per pound of sulfur

$100 per pound of lead removed

We know that the net cost can be written as,

T(x, y) = 7,000 + 100x² + 50y² -500x -100y

where x and y is the amount of sulfur and lead emission reduced respectively.

To find the minimum of x differentiate the value of net cost with respect to x,

[tex]\dfrac{dT}{dx} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}[/tex]

    [tex]= \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}\\\\= 0+ 100(2x) + 0 + 500 + 0\\\\ = 200x+500[/tex]

Substitute against 0, to get the minimum value of x,

0 = 200x+500

x = 2.5

Differentiate again,

[tex]\dfrac{d^2T}{dx^2} = \dfrac{d(200x+500)}{dx}[/tex]

      [tex]=200+0[/tex]

As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.

To find the minimum of y differentiate the net cost with respect to y,

[tex]\dfrac{dT}{dy} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dy}[/tex]

     [tex]= 0+0+50(2y)-100\\\\=100y-100\\\\=100(y-1)[/tex]

Substitute against 0 to get the minimum value of y,

0 = 100(y-1)

y = 1

Differentiate again,

[tex]\dfrac{d^2T}{dy^2} = \dfrac{d(100y-100)}{dy}[/tex]

       [tex]=100[/tex]

As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.

Hence, the amount of sulfur and lead the firm should remove is 2.5 pounds and 1 pound respectively to minimize the net cost.

Learn more about Differentiation:

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Answer: m= $25/10 r

Step-by-step explanation:

Let m= money

r= raffle ticket

Then according to the statement

m= $25 for 10 tickets

so 10 tickets= $25

Or the equation goes,

m= $25/10 r

Answer:

44.4%

Step-by-step explanation:

Additional amount they were adding to equipment cost is $900

Percentage of $900 that is $400 = $400/$900 × 100 = 44.4%

$400 is approximately 44.44% of the additional $900 being added to the equipment cost.

To calculate what percent $400 is out of the additional $900 being added to the equipment cost, we can use the following formula:

In this case, the part is $400, which is the predicted additional profit from advertising, and the whole is the total additional amount of $900 being added to the equipment cost.

Using the formula:

So, $400 is 44.44% (approximately) of the additional $900 that the brothers were planning to add to the equipment cost.

Answer: the salesperson earns $54 as commission.

Step-by-step explanation:

A furniture salesperson earns 4.5% commission on every piece of furniture sold. The salesperson sells a sofa for $1000. This means that the commission that he earned from the sale of the sofa is

4.5/100 × 1000 = 0.045 × 1000 = $45

The salesperson also sold a chair for $200. This means that the commission that he earned from the sale of the chair is

4.5/100 × 200 = 0.045 × 200 = $9

The total commission that the salesperson earns is

45 + 9 = $54

The question is incomplete. The map is attached as a photo and here is the complete question:

a. What is the actual distance between Saugerties and

Kingston? ___________________________

b. Catskill is 15 miles from Saugerties. What would the

distance on the map be? ___________________________

c. On another map, the distance between Saugerties and

Kingston is 2 inches. What would the distance from Saugerties to

Catskill be on this map? __________________

Since you have asked the answer for part (c) only, here is the answer:

Answer:

The distance from Saugerties to Catskill would be 3 inches on this map.

Step-by-step explanation:

On the given map, the distance between Saugerties and Kingston is 4 inches and the scale is 1 inch = 2.5 miles.

To calculate the actual distance in miles between Saugerties and Kingston, consider the given scale:

1 inch ---------- 2.5 miles

4 inches ------ x miles

Cross multiplying:

1x = 2.5 x 4

x = 10 miles

The actual distance between Saugerties and Kingston is 10 miles.

On another map, the distance between Saugerties and Kingston is 2 inches. Which means the scale is 2 inches = 10 miles. So 1 inch = 10/2 = 5 miles

The scale on the other map is 1 inch = 5 miles.

Catskill is 15 miles from Saugerties (given in the previous part). So on the map:

1 inch -------- 5 miles

y inch ------- 15 miles

Cross multiplying:

15 = 5y

y = 15/5

y = 3 inches.

The distance from Saugerties to Catskill would be 3 inches on this map.

Answer:

True

Step-by-step explanation:

Statistics is indeed a procedure consists of collection, presentation, analysis and presentation of the data. Statistics is widely used in every field of life. For example we want to assess the performance of students. For this purpose we choose a sample of students and ask about their grading. This is the procedure of collection of data. Then this data can be presented in tabular form or graphical form and this stage is known as presentation of the data. Then we evaluate their average and this procedure is known as analysis of data. Then according to average we make statement about performance of students and this stage is interpretation of the data.

Answer:

Measurement of all angle= [tex]76.79\º+102.66\º+15.35\º+265.16\º= 360\º[/tex]

Step-by-step explanation:

Given angles are [tex]x\º, (\frac{5x}{9} +60)\º, (\frac{x}{5} )\º, (4x-142)\º[/tex]

The given is a quardilateral.

We know the sum of all angles of quardilaterals is 360º

∴ [tex]x\º+ (\frac{5x}{9} +60)\º+ (\frac{x}{5} )\º+(4x-142)\º= 360\º[/tex]

Now, solving the equation to find value of x.

⇒ [tex]x\º+ (\frac{5x}{9} +60)\º+ (\frac{x}{5} )\º+(4x-142)\º= 360\º[/tex]

Opening parenthesis.

⇒ [tex]x\º+ \frac{5x}{9} +60\º+ \frac{x}{5} \º+4x-142\º= 360\º[/tex]

⇒ [tex]5x\º+ \frac{5x}{9} + \frac{x}{5} \º-82\º= 360\º[/tex]

Adding both side by 82

⇒ [tex]5x+ \frac{5x}{9} + \frac{x}{5} = 442[/tex]

Taking LCD 45

⇒ [tex]\frac{45\times 5x+ 5\times 5x+9x}{45} = 442[/tex]

Multiplying both side by 45

⇒ [tex]225x+25x+9x= 19890[/tex]

⇒[tex]259x= 19890[/tex]

Dividing both side by 259

⇒[tex]x= \frac{19890}{259}[/tex]

∴[tex]x= 76.79\º[/tex]

Next subtituting the value of x to find measurement of other interior angle.

[tex]x\º, (\frac{5x}{9} +60)\º, (\frac{x}{5} )\º, (4x-142)\º[/tex]

2. [tex](\frac{5x}{9} +60)\º[/tex]

= [tex]\frac{5\times 76.79}{9} +60= 42.66+ 60[/tex]

= [tex]102.66\º[/tex]

3. [tex](\frac{x}{5} )\º[/tex]

= [tex](\frac{76.79}{5} )\º= 15.35\º[/tex]

4. [tex](4x-142)\º[/tex]

= [tex]4\times 76.79- 42= 307.16-42[/tex]

= [tex]265.16\º[/tex]

Answer:

  78

Step-by-step explanation:

You want to know the sum of two 2-digit numbers such that one differs from the other by a multiple of 10, and the difference of their squares is 2340.

The difference of squares is the product of the sum and difference of the two numbers. Here, the difference must be a multiple of 10, so we want factorizations of 2340 that have a multiple of 10 as a factor. These are ...

  2340 = 10(234) = 20(117) = 30(78)

where the first factor (the difference) is less than the second factor (the sum).

The sum of two 2-digit numbers cannot be more than 200, and it must be even if they both have the same units digit. The only viable product from the above list is 30 × 78, where 30 is the difference of the numbers and 78 is their sum.

The sum of the numbers is 78.

__

Additional comment

The two numbers are 54 and 24. The difference of their squares is ...

  2916 -576 = 2340

Difference of squares: a² -b² = (a -b)(a +b).

Answer:

option B

Step-by-step explanation:

The described data is qualitative because regions of a residence has categories and it don't measure or count anything. The regions of residence can be divided into different categories and numerical quantities can be assigned to them but still they will be qualitative variables because numerical quantities will be working as identifiers and they don't measure or count anything.

Answer:

one penny = S

one nickel = T

one dime = U

one quater = V

∴ Probability one obtaining at least one of S, T, U or V = 4/24 = 1/6

Step-by-step explanation: They can be arranged as follows

1) STUV    

2)TSUV

3) TSVU

4) UVST

5) VUST

6) VUTS

7) TVUS

8) TVSU

9) TUVS

10) SVUT

11) VSTU

12) VSUT

13) SVTU

14) UTSV

15) UTVS

16) USTV

17) USVT

18) VTUS

19) VTSU

20) SUTV

21) SUVT

22) STVU

23) TUSV

24) UVTS

Answer:

Step-by-step explanation:

A yard of lace costs w cents a yard and fabric costs $.40 more than the lace. This means that the cost of a yard of fabric would be

w + 0.5

Kimberly wants to buy one yard of lace and 2 yards of fabric. This means that the total amount of money that she would have to pay for the lace is is

0.4 × 1 = 0.40

Amount that she would spend on the fabric is 2(w + 0.5) = 2w + 1

Total cost would be

0.4 + 2w + 1

Final answer:

Kimberly will need a total of 3w + 80 cents to purchase one yard of lace and two yards of fabric, where w is the cost of one yard of lace in cents.

Explanation:

To calculate how much money Kimberly will need to buy one yard of lace and two yards of fabric, first we identify the cost of one yard of lace as w cents. The fabric costs $0.40 more than the lace per yard, so the cost of one yard of fabric is w cents + 40 cents. Kimberly wants to buy two yards of fabric, so we multiply the cost of one yard of fabric by 2, which gives us 2(w + 40) cents.

Adding together the cost of the lace and the two yards of fabric, we get: w + 2(w + 40). Simplifying this expression, we have: w (cost of one yard of lace) + 2w (twice the cost of lace per yard for two yards of fabric) + 80 (twice the additional cost of fabric per yard)

w + 2w + 80 cents

3w + 80 cents

Therefore, Kimberly will need a total of 3w + 80 cents to purchase one yard of lace and two yards of fabric.

n = the number of miles traveled

0.75n + 1.75 ≤ 20     Option D

[costs $0.75 for each additional mile (n) plus $1.75 for the first mile less than or equal to $20 (because you have a maximum of $20 and you can't spend more than that)]

The inequality to determine the number of additional miles you can travel with $20, considering a base fare of $1.75 and $0.75 per additional mile, is 0.75n + 1.75 ≤ 20.

The subject of this question is to determine which inequality represents the situation of a taxi cab fare and how many miles can be traveled with $20 given the cost per mile. The correct inequality is the one that starts with the base fare and adds the cost for each additional mile multiplied by the number of miles. The base fare is $1.75 for the first mile and there is an additional cost of $0.75 for each additional mile.

Let n represent the number of additional miles you can travel. Since you have $20, the inequality to find out how many miles you can travel would be the money spent on additional miles plus the base fare has to be less than or equal to $20 you have. This is represented by the inequality 0.75n + 1.75 ≤ 20. This means we are adding $0.75 for each additional mile (n) to the base fare of $1.75, and the total cost must not exceed $20 to stay within the budget.

Answer: d

Step-by-step explanation: Both country A and B has equal number of workers,physical capital, human capital and access to natural resources.

Non is higher than the other.

Each chef at "sushi emperor" prepares 15 regular rolls and 20 vegetarian rolls daily. On tuesday, each customer ate 2 regular rolls & 3 vegetarian rolls. by the end of the day, 4 regular rolls & 1 vegetarian roll remained uneating. how many chefs were on tuesday ? and how many customers were they ?

There were 2 chefs and 13 customers on tuesday

Let x be the number of chefs at Sushi Emperor and y be the number of customers on Tuesday.

From given,

Each chef prepares 15 regular rolls and 20 vegetarian rolls daily

If each chef prepares 15 regular rolls, then x chefs prepare 15x regular rolls

If each customer ate 2 regular rolls, then y customers ate 2y regular rolls

By the end of the day, 4 regular roll remained un eating

Therefore,

15x - 2y = 4 --------- eqn 1

If each chef prepares 20 vegetarian rolls, then x chefs prepare 20x vegetarian rolls

If each customer ate 3 vegetarian rolls, then y customers ate 3y vegetarian rolls

By the end of the day, 1 vegetarian roll remained uneating

Therefore,

20x - 3y = 1 ---------- eqn 2

Let us solve eqn 1 and eqn 2

Multiply eqn 1 by 3

45x - 6y = 12 ------- eqn 3

Multiply eqn 2 by 2

40x - 6y = 2 ------- eqn 4

Subtract eqn 4 from eqn 3

45x - 6y = 12

40x - 6y = 2

( - ) --------------

5x = 10

x = 2

Substitute x = 2 in eqn 1

20(2) - 3y = 1

40 - 3y = 1

3y = 39

y = 13

Thus there were 2 chefs and 13 customers

Answer: Paul is 18 years old and Gary is 12 years old.

Step-by-step explanation:

Let x represent the age of Paul.

Paul is six years older than Gary. This means that Gary's age would be

x - 6

The sum of their ages is 30. This means that

x + x - 6 = 30

2x - 6 = 30

Adding 6 to the left hand side and the right hand side of the equation, it becomes

2x - 6 + 6 = 30 + 6

2x = 36

Dividing the left hand side and the right hand side of the equation by 2, it becomes

2x/2 = 36/2

x = 18

Therefore, Paul is 18 years old.

Gary is 18 - 6 = 12 years old.

Final answer:

Rosa saves 30% of her weekly earnings for college, which is calculated by dividing the amount saved ($36) by her total weekly earnings ($120) and then multiplying by 100%.

Explanation:

To determine the percentage of her earnings that Rosa saves, we will use the formula for calculating percentage: Percentage saved = (Amount saved ÷ Total earnings) × 100%.

First, we identify the total earnings and the amount saved: Rosa's total weekly earnings are $120, and she saves $36 each week.

Next, we calculate the percentage: Percentage saved = ($36 ÷ $120) × 100% = 0.3 × 100% = 30%.

Therefore, Rosa saves 30% of her weekly earnings for college.

Answer:

There are 7 dogs.

Step-by-step explanation:

7 (dogs) x 7 (biscuits) = 49

3 (cats) x 6 (biscuits) = 18

49 + 18 = 67

To calculate the time spent above 40 meters on a Ferris wheel, we analyze the wheel's structure and use trigonometry to find the proportional time of the ride corresponding to the arc above the 40-meter mark.

Calculating Time Spent Above 40 meters on a Ferris Wheel

To determine how many minutes of the ride on the Ferris wheel are spent higher than 40 meters above the ground, we need first to understand the wheel's structure and motion. Given that the Ferris wheel has a diameter of 50 meters and is boarded from a platform 4 meters above the ground, we can calculate the highest and lowest points riders will reach during the ride. The highest point of the wheel will be 50 meters (the radius) plus 4 meters (the platform height), totaling 54 meters. The lowest point will be 4 meters (platform height) above the ground since the diameter is equal to twice the radius and the wheel's lowest point touches the platform level.

Because riders want to be above 40 meters, we're interested in the segment of the ride where they are between 40 meters and the maximum of 54 meters above the ground. This range covers a portion of the wheel's circumference. If the wheel's highest point is at 12 o'clock and the loading platform is at 6 o'clock, then the 40-meter height will be somewhere between 6 o'clock and 12 o'clock.

Using the fact that the Ferris wheel completes one revolution in 4 minutes, we can find the time spent over 40 meters by calculating the angle θ that corresponds to the arc between 40 meters and 54 meters above the ground. The 40-meter height will fall at the point where the vertical distance from the top of the wheel equals the wheel's radius minus 40 meters. Using this, we can use trigonometry to find θ, and then convert this angle to a proportion of the full revolution time to determine the time spent on this segment of the ride.

The question is about identifying a reaction mechanism based on the provided rate law, which is derived from the reactant concentrations. A likely mechanism is a single-step process where the reactants come together to form the product, mirroring the rate law's implication that reactant concentration directly influences the reaction rate.

The student is asking about the mechanism of a reaction and its rate law. In chemistry, the rate law expresses the relationship among the reaction rate, the reaction mechanism, and the concentrations of the reactants.

The provided rate law is rate = k[x2y2][z2], which suggests that the reaction is first order with respect to both x2y2 and z2. A plausible mechanism for this reaction would involve a single step where one molecule of each reactant comes together to form the product, 2x2y2z.

In a single-step mechanism, the rate law is derived directly from the stoichiometry of the reactants in the balanced chemical equation. Since the rate law matches the stoichiometry of the reactants, it is likely that this could represent an elementary reaction, given that the stoichiometry and the rate law coincide.

The direct relationship implied by the rate law between the concentration of the reactants and the reaction rate can be explained by the likelihood of reactant particles colliding and reacting being greater with increased concentrations, a concept fundamental in kinetics.

Answer:

The answer to your question is the probability to select a yogurt of raspberry is 1/4 or 25%.

Step-by-step explanation:

Data

Number of cartons = 12

Number of cartons of raspberry = 3

To solve this problem, just use the formula of probability and simplify it to get the result.

Formula

P(A) = [tex]\frac{Number of favorable outcomes to A}{Total number of outcomes}[/tex]

Substitution

P(A) = 3/12

Simplification

P(A) = 1/4 or 25%

Answer:

Only one person is in the stopped vehicle

Step-by-step explanation:

To solve the programming task, you would create a loop to read integers, use a counter to track consecutive duplicates and print the count after a negative number is entered to end the input.

The student's question involves writing a piece of code that reads a sequence of non-negative integers, ends the input with a negative integer, and reports the number of consecutive duplicate values entered before the negative integer is encountered. This is a programming task that typically involves a loop and a counter.

To address this question, you would write a loop that continues to accept input until a negative number is entered. Inside the loop, you would use a counter to keep track of the number of consecutive duplicates. Here is a pseudocode example:

Initialize the previous Value to None (or some value that won't occur in the sequence).

Initialize the count Of Duplicates to 0.

Start a loop that reads integers until a negative number is encountered.

Inside the loop, compare the current number to the previous Value.

If they are the same, increment the count Of Duplicates by 1.

Set the previous Value to the current number before the next iteration.

Outside the loop, print count Of Duplicates.

The counter pattern is a fundamental concept in programming that is used to tally occurrences within iteration structures like loops.

Answer:

The correct answer is C. 30 = 9x + 7

Correct statement and question:

Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of x feet each.

Answer choices:

A. 7x + 9 = 30

B. 30x + 7 = 9

C. 30 = 9x + 7

D. 9x - 7 = 30

Source:

https://quizizz.com/admin/quiz/5c94aa300d3459001a4ef259/unit-6-lesson-4-equations-and-word-problems

Step-by-step explanation:

1. Information given to us to answer the problem correctly:

Length of the piece of ribbon for a craft project = 30 feet

First cut = 7 feet

Remaining piece cut into 9 equal lengths of x feet each

2. Let's find the right equation to solve for x:

9x + 7 = 30

The nine equal pieces of x feet each plus the piece of 7 feet add up to 30 feet

The correct answer is C. 30 = 9x + 7

Yo sup??

y=-3/x+4

cross multiply

x+4=-3/y

x=-3/y-4

f(y)=-3/y-4

or

f(x)=-3/x-4

=-4x-3/x

The correct answer is option 4

Hope this helps.

Answer:

2038000 visitors

Step-by-step explanation:

Now in question expression has been repeated, so original expression is

[tex]-8.5x^2+103x+2026[/tex]

Now according to given information "x" represent the number of years after 2006. So, for 2018, it will be

2018-2006 = 12 years

So, x = 12

putting value of x in expression

(-8.5)x(12)^2 + (103)x(12) + 2026

-1224 + 1236 + 2026

2038

Now the model shows the number of visitors in thousand, so you have to multiply your answer in thousand

2038 x 1000

= 2038000 visitors

Answer:

$88812.33

Step-by-step explanation:

Principal = $89,000

But there's a sales tax of 4.2%

Tax = (4.2 / 100) * 89000 = $3738

Total cost of the mobile house = $89000 + $3738 = $92,738

However, they made a down payment of $3000

Balance after down payment = $92,738 - $3000 = $89,738

They've also paid the first monthly payment of $925.67

Balance after first monthly payment =

$89,738 - $925.67 = $88,812.33

The principal balance after applying there first monthly payment is $88,812.33

The principal balance, after applying the first month's payment of $925.67 to an initial loan of $89,738 for a mobile home, is $88,812.33.

To calculate the principal balance after the first month’s payment, start by adding the initial price of the mobile home and the sales tax. The sales tax is 4.2% (or 0.042) of $89,000, which comes out to $3,738. Then, subtract the down payment from the sum to get the initial loan amount. So, $89,000 + $3,738 - $3,000 equals $89,738. After applying the first month’s payment of $925.67, subtract that amount from the initial loan, leaving a principal balance of $88,812.33.

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Answer: 61.9°

Approximately: 62°

Step-by-step explanation:

Using Pythagoras

(AC)^2 = 17^2 - 15^2

= 289 - 225

√AC = √64

AC = 8

Cos C = Adj/ Hyp

Cos C = 8/17

Cos C = 0.4706

Inverse of Cos C

C = 61.93°

Approximately: 62°

The average rate of change for the function over the interval 1 ≤ x ≤ 3 is 2.

The average rate of change for a function can be calculated by finding the difference in the function values over the given interval and dividing it by the difference in the x-values over the same interval.

For the given function graphed, the average rate of change over the interval 1 ≤ x ≤ 3 is calculated as follows:
Average rate of change = (f(3) - f(1)) / (3 - 1)

By evaluating the function at x = 1 and x = 3 and applying the formula, we find:
Average rate of change = (6 - 2) / 2 = 4 / 2 = 2

Therefore, the BEST estimate of the average rate of change for the function over the interval 1 ≤ x ≤ 3 is 2 (option A).

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Answer:

d

Step-by-step explanation:

The probability of computers are 20%. Therefore to determine the amount that would break out of 10 we simply just multiply by 0.2.

[tex]=10\cdot{0.2}=2[/tex]

Therefore at most two breaks down. The answer is d

Answer:

Step-by-step explanation:

I'm going to use calculus to solve this, because it's the simplest way.  

The acceleration due to gravity in feet is the second derivative of the position function.  We will start with the acceleration and work backwards with antiderivatives to get to the position function.

a(t) = -32.  Going backwards and using the fact that the initial vertical velocity is 64 ft/sec, our velocity function is

v(t) = -32t + 64.  Going backwards and using the fact that the initial height of the ball is 6 feet, our position function is

[tex]s(t)=-16t^2+64t+6[/tex]

The first part of this question asks us the maximum height of the ball.  From Physics, we learn that the maximum height of a projectile is reached when the velocity is 0, which happens to be right where the projectile stops for a nanosecond in the air to turn around and come back down.  We set the velocity function equal to 0 and solve for t.

0 = -32t + 64 and

0 = -32(t - 2).  By the Zero Product Property, either -32 = 0 or t - 2 = 0.  It's obvious that -32 does not equal 0, so t - 2 must equal 0.  Solving this for t:

t - 2 = 0 so

t = 2 seconds.  Since the maximum height is reached at a time of 2 seconds, we plug 2 seconds into the position function to get its position at 2 seconds (which is also the max height of the ball).

[tex]s(2)=-16(2)^2+64(2)+6[/tex] and

s(2) = -64 + 128 + 6 so

s(2) = 70 feet

Now we want to know when the ball will hit the ground.  "When" is a time value, and we know that the height of the ball on the ground is 0, so we sub in a 0 for s(t) and factor the quadratic.

Using the quadratic formula:

[tex]t=\frac{-64+/-\sqrt{4096-4(-16)(6)} }{-32}[/tex] and

[tex]t=\frac{-64+/-\sqrt{4480} }{-32}[/tex] which gives us the 2 solutions

[tex]t=\frac{-64+\sqrt{4480} }{-32}[/tex] and

[tex]t=\frac{-64-\sqrt{4480} }{-32}[/tex]

Plugging into your calculator, the first t = -.0916500 and the second t = 4.091

We all know that time cannot ever be negative, so our t value is 4.09.

Again, from Physics, we know that a projectile reaches it max height at halfway through its travels, so it just goes to follow logically that if it halfway through its travels at 2 seconds, then it will hit the ground at 4 seconds.  And it does!! How awesome is that?!

The maximum height of the ball is 32 feet. The ball will hit the ground after 4 seconds.

To find the maximum height of the ball, we can use the equation for motion under constant acceleration.

The equation is h = (v0^2)/(2g),

where h is the maximum height, v0 is the initial vertical velocity, and g is the acceleration due to gravity.

Plugging in the given values, we have h = (64^2)/(2*32), which gives us a maximum height of 32 feet.

Next, to find when the ball will hit the ground, we can use the equation for time of flight.

The equation is t = (2v0)/g, where t is the time of flight.

Plugging in the given values, we have t = (2*64)/32, which gives us a time of flight of 4 seconds.

https://brainly.com/question/20627626

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