20 points!!

The coordinates of the vertices of quadrilateral JKLM are J(−3, 2) , K(3, 5) , L(9, −1) , and M(2, −3) .


Which statement correctly describes whether quadrilateral JKLM is a rhombus?
Quadrilateral JKLM is not a rhombus because there is only one pair of opposite sides that are parallel.

Quadrilateral JKLM is a rhombus because opposite sides are parallel and all four sides have the same length.

Quadrilateral JKLM is not a rhombus because there are no pairs of parallel sides.

Quadrilateral JKLM is not a rhombus because opposite sides are parallel but the four sides do not all have the same length

Answers

Answer 1
Final answer:

Using the distance formula, we can calculate the distances between the vertices of quadrilateral JKLM. After performing these calculations, we can determine that not all sides have the same length, thus quadrilateral JKLM is not a rhombus.

Explanation:

To determine if quadrilateral JKLM is a rhombus, we need to check if all four sides are of equal length because a defining property of a rhombus is that it has all sides of equal length. The length between two points (x1, y1) and (x2, y2) can be found using the distance formula which is √[(x2-x1)² + (y2-y1)²].

After calculating, we find:
J-K distance = √[(5-2)² + (-3-3)²] = √42
K-L distance = √[(-1-5)² + (9-3)²] = √52
L-M distance = √[(-3- -1)² + (2-9)²] = √52
M-J distance = √[(2- -3)² + (-3-2)²] = √42

Therefore, the distance between vertices J-K and L-M are the same, and the distance between vertices K-L and M-J are the same, but the two pairs are not in themselves equal to each other. As such, quadrilateral JKLM is not a rhombus because all sides do not have the same length.

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Related Questions

which overlapping triangles are congruent by ASa

Answers

Depends where ASa  is if we have a picture it would help a little bit thx

Jacqui has grades of 79 and 76 on her first two algebra tests. if she wants an average of at least 71​, what possible scores can she make on her third​ test?

Answers

Hello There!

79 + 76 + x = ?
71 x 3 = 213
79 + 76 =  155
213 - 155 = 58
x = 58
She can score anything 58 or above to acquire her minimum average of 71.

Hope This Helps You!
Good Luck :) 

- Hannah ❤

PLEASE HELP MATH!!!!!!

Answers

The slope of the line segmenThe slope of the line segment would be 1/4 because the graph is rising by 4 and running by 1, hence 1/4.
1. (1,4)(2,8)
     slope = (8 - 4) / (2 - 1) = 4 <==
=================
2. Brand A : 9 * 12 = 108
    Brand B : 9 * 15 = 135
   so Brand B costs (135 - 108) = 27 bucks more
=================
3. both calculated 2/9
=================
4. initial value is 5
    

What is the value of n in the numerical sentence below?

√16 ÷ 4³ = 4ⁿ

n = ?

Answers

n= -2 because square root of 16 is 4 and 4 is raised to the power 1. The bases are the same so you subtract the exponents since it's division
n = -2
see attachment for calculation.

Triangle LMN is located at L(2,3), M(1,2), and N(4,4). The triangle is then transformed using the rule (x+5, y+2) to form the image L’M’N. What are the new coordinates of L’,M’, and N’?

Answers

using the rule (x+5, y+2) you would add 5 to the x value and 2 to the y value so the answer would be L' (7,5) M' (6,4) N' (9,6)

Given:- L (2, 3), M (1, 2), and N make up the triangle known as LMN (4, 4). The image L'M'N' is created by first transforming the triangle according to the rule

To Locate:

What are L's, M's, and N's new coordinates?

Solution:-

Rule for Points = (x + 2) on the x-axis.

Thus, the new x-points axis's are as follows:

L' = 2 + 5 = 7

M' = 1 + 5 = 6

N' = 4 + 5 = 9 .

and, Rule for y - axis Points = (y + 2) .

Therefore, New y - axis points of new ∆ are :-

L' = 3 + 2 = 5

M' = 2 + 2 = 4

N' = 4 + 2 = 6 .

Hence, New coordinates of L', M', and N' :-

L' = (7 , 5)

M' = (6, 4)

N' = (9, 6)

Find out more:

show that the point of trisection for the line connecting the points (-5,12) and (-1,12) is the midpoint of the line connecting the...

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the total cost of a bus ride and a ferry ride is $8.00. in one month, bus fare will increase by 10% and ferry fare will increase by 25%. the total cost will then be 9.25. how much is the current bus fare?

Answers

The bus fare is x, and the ferry fare is y.

x + y = 8

The bus fare will increase by 10% and the ferry fare will increase by 25%.

1.1x + 1.25y = 9.25

Multiply the first equation by -1.1:

-1.1x - 1.1y = -8.8

Add this to the original second equation.

0.15y = 0.45

y = 3

x + y = 8

x + 3 = 8

x = 5

The bus fare is $5, and ferry fare is $3.

A scuba diver descends at a rate of 40 feet per minute. How many feet will the scuba diver move in 2 minutes?

Answers

80 feet! 

40 feet x 2 minutes = 80 feet 

In response to the increasing weight of airline passengers, the Federal Aviation Administration in 2003 told airlines to assume that passengers average 189.7 pounds in the summer, including clothing and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 41.4 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 27 passengers.


What is the approximate probability (±0.0001) that the total weight of the passengers exceeds 5511 pounds?

Answers

The only way I can think of to solve this problem is to assume normal distribution.

Since the total weight excess 5511, hence the weight per passenger is at least:

x = 5511 / 27 = 204.1 pounds

 

Solve for the z score:

z = (x – u) / s

z = (204.1 – 189.7) / 41.4

z = 0.35

 

From the standard probability tables, the P value using right tailed test is:

P = 0.3632

Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of the quantity 2 times x to the 4th power plus 8 times x and 4 times x to the 3rd power plus 4, dx. Your work must include the use of substitution and the antiderivative.

Answers

[tex]\bf \displaystyle \int_{-1}^{0}~(2x^4+8x)^3(4x^3+4)\cdot dx\\\\ -------------------------------\\\\ u=2x^4+8x\implies \cfrac{du}{dx}=8x^3+8\implies \cfrac{du}{2(4x^3+4)}=dx\\\\ -------------------------------\\\\ \displaystyle \int_{-1}^{0}~u^3\underline{(4x^3+4)}\cdot \cfrac{du}{2\underline{(4x^3+4)}}\implies \cfrac{1}{2}\int_{-1}^0~u^3\cdot du\\\\ -------------------------------\\\\ [/tex]

[tex]\bf \textit{now, let's change the bounds, using }u(x)\\\\ u(-1)=2(-1)^4+8(-1)\implies u(-1)=-6 \\\\\\ u(0)=2(0)^4+8(0)\implies u(0)=0\\\\ -------------------------------\\\\ \displaystyle \cfrac{1}{2}\int_{-6}^0~u^3\cdot du\implies \left.\cfrac{1}{2}\cdot \cfrac{u^4}{4} \right]_{-6}^0\implies \left. \cfrac{u^4}{8} \right]_{-6}^0 \\\\\\\ [0]~~-~~[162]\implies -162[/tex]

notice, by changing the bounds using u(x), we do not need to substitute back the "u".

Answer:

-162

Step-by-step explanation:

Integration by substitution is a technique in calculus where a change of variables is made to simplify the integral by expressing it in terms of a new variable, making the integration easier to manage.

To evaluate the following integral, use the method of substitution:

[tex]\displaystyle \int_{-1}^0 (2x^4 + 8x)^3 (4x^3 + 4) \;\text{d}x[/tex]

Let u = 2x⁴ + 8x.

Differentiate u with respect to x using the power rule for differentiation by multiplying each term by its exponent and then subtracting 1 from the exponent:

[tex]\dfrac{\text{d}u}{\text{d}x}=4 \cdot 2x^{4-1}+1\cdot 8 x^{1-1}[/tex]

[tex]\dfrac{\text{d}u}{\text{d}x}=8x^{3}+8[/tex]

Rewrite it so that dx is on its own:

[tex]\text{d}u=8x^{3}+8\;\text{d}x[/tex]

[tex]\text{d}x=\dfrac{1}{8x^{3}+8}\;\text{d}u[/tex]

Change the limits of integration from x to u:

[tex]\begin{aligned}x=-1 \implies u&=2(-1)^4 + 8(-1)\\&=2(1)-8\\&=2-8\\&=-6\end{aligned}[/tex]

[tex]\begin{aligned}x=0 \implies u&=2(0)^4 + 8(0)\\&=2(0)-0\\&=0-0\\&=0\end{aligned}[/tex]

Rewrite the integral in terms of u:

[tex]\begin{aligned}\displaystyle \int_{-1}^0 (2x^4 + 8x)^3 (4x^3 + 4) \;\text{d}x&=\int_{-6}^0 u^3 (4x^3 + 4) \cdot \dfrac{1}{8x^{3}+8}\;\text{d}u\\\\&=\int_{-6}^0 u^3 (4x^3 + 4) \cdot \dfrac{1}{2(4x^{3}+4)}\;\text{d}u\\\\&=\int_{-6}^0\dfrac{u^3}{2} \;\text{d}u\end{aligned}[/tex]

Integrate with respect to u using the power rule for integration by adding 1 to the exponent of each term and then dividing by the new exponent:

[tex]\begin{aligned}\displaystyle \int_{-6}^0\dfrac{u^3}{2} &=\left[\dfrac{u^{3+1}}{2 \cdot (3+1)}\right]^0_{-6}\\\\&=\left[\dfrac{u^{4}}{8}\right]^0_{-6}\\\\&=\dfrac{(0)^4}{8}-\dfrac{(-6)^4}{8}\\\\&=0-\dfrac{1296}{8}\\\\&=-162\end{aligned}[/tex]

Therefore, the value of the given integral is -162.

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form. (1 point) 4, -14, and 5 + 8i

Answers

since it is a polynomial
for the zero 4 you can write (x-4)
for the zero -14 you can write (x+14)
for the zero 5+8i since it is complex it will be accompanied with its conjugate 5-8i so you can write (x-(5+8i) and (x-(5-8i)) =(x^2-10x+89)
so
(x-4)(x+14)(x^2-10x+89)
expanding
x^4-67x^2+1450x-4984=0

The required polynomial function is x⁴ - 67x² + 1450x - 4984 = 0 with 4 degrees with real coefficients.

What is a polynomial?

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.

The zeros are given in the question as 4, -14, and 5 + 8i

The required polynomial function of minimum degree with real coefficients whose zeros include those listed above.

For the zero 4, you can write (x-4)

For the zero -14, you can write (x+14)

For the zero 5+8i since it is complex it will be accompanied by its conjugate 5-8i

So, you can write (x-(5+8i) and (x-(5-8i)) =(x²-10x+89)

(x - 4)(x + 14)(x² - 10x + 89)

Expanding the expression, we get

x⁴ - 67x² + 1450x - 4984 = 0

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Write what the expressions below best represent within the context of the word problem

How many quarts of water must be added to 4 quarts of an 80% mixture to obtain a 50% mixture?

x represents quarts of water to be added to the ____% mixture.
x + 4 represents the total quarts of the mixture at ____%.

Answers

first one is 80%

second one is 50%

1st. is 80%

2nd. is 50%

Danielle can spend up to $600 buying small and large jackets for her employees. Small jackets cost $20 each and large jackets cost $25 each. To get a special discount, she needs the number of large jackets to be no more than twice the number of small jackets. Also, she needs to order at least 6 small jackets and 10 large jackets. The graph shows the feasible region, where x represents the number of small jackets and y represents the number of large jackets. Which ordered pairs meet all the constraints and make sense in context of the situation? Select each correct answer. (13, 11) (15, 11) (8, 9) (7, 15) (10,13)

Answers

Answer:

10,13 13, 15,11

Step-by-step explanation:

THANK YOU PREVIOUS PERSON YOU SAVED MY WHOLE MATH GRADE! I OWE YOU  YOU MADE IT SO I CAN PASS 9TH GRADE.

Find the solution of the square root of the quantity of x plus 3 plus 4 equals 6, and determine if it is an extraneous solution

Answers

x=1 non extraneous i hope this helps

Answer:

[tex]x=1[/tex]

Step-by-step explanation:

We have been given an equation [tex]\sqrt{x+3}+4=6[/tex]. We are asked to find the solution of our given equation.

[tex]\sqrt{x+3}+4-4=6-4[/tex]

[tex]\sqrt{x+3}=2[/tex]

Now, we will square both sides of our given equation.

[tex]x+3=2^2[/tex]

[tex]x+3=4[/tex]

[tex]x+3-3=4-3[/tex]

[tex]x=1[/tex]

To see whether [tex]x=1[/tex] is an extraneous solution or not, we will substitute [tex]x=1[/tex] in our given equation as:

[tex]\sqrt{1+3}+4=6[/tex]

[tex]\sqrt{4}+4=6[/tex]

[tex]2+4=6[/tex]

[tex]6=6[/tex]

Since both sides of our given equation are equal, therefore, [tex]x=1[/tex] is a solution for our given equation.

A division of a company produces income tax apps for smartphones. each income tax app sells for $9. the monthly fixed costs incurred by the division are $25,000, and the variable cost of producing each income tax app is $4. (a) find the break-even point for the division.

Answers

The equation for profit is income – cost:

Profit = Income – Cost

Let us say that x is the number of sold amount

 

Profit = 9 x – (25,000 + 4 x)

Profit = 5 x – 25,000

 

Breakeven point occurs when Profit = 0, hence:

5 x = 25,000

x = 5,000

 

The breakeven is when 5,000 people uses the income tax app




Ricardo is constructing a line through point P that is perpendicular to line m. He has already constructed the arc shown.
He places his compass on point X to construct an arc.

What must be true about the width of the compass opening when Ricardo draws the arc?

It must be less than 1/2XY.

It must be equal to XY .

It must be equal to PX .

It must be greater than 12XY.

Answers

Answer:

It must be greater than 1/2XY.

Step-by-step explanation:

We draw an arc from point X below the arc that passes through X and Y.  Keeping our compass the same width, we will draw another arc from point Y below, intersecting the first arc.

If the width of the compass is not set to more than 1/2XY, then the two arcs will not intersect and we will not complete our construction.

Perpendicular lines are lines that meet at 90 degrees.

The true statement about the width of the compass is that: (d) It must be greater than 1/2XY.

From the question, we understand that he has drawn arc XY already.

The next step is to draw an arc less than the width of XY, but greater than half width XY.

This will ensure that the arcs bisect one another.

Hence, the true option is (d).

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Kyra is using rectangular tiles of two types for a floor design. A tile of each type is shown below: Two rectangular tiles, rectangle PQRS with vertices at P 1, 1. Q is at 8, 1. R is at 8, 5. S is at 1, 5. Rectangle JKLM with vertices J at 4, 1. K is at 8, 1. L is at 8, 4. M is at 4, 4. Which statement is correct? The two tiles are not similar because segment SP is to segment SR is 4:7 and segment MJ is to segment ML is 1:3. The two tiles are similar because segment PQ is to segment QR is 4:3 and segment JK is to segment KL is also 4:3. The two tiles are similar because segment SR is to segment ML is 7:4 and segment PQ is to segment JK is also 7:4. The two tiles are not similar because segment PQ is to segment QR is 7:4 and segment JK is to segment KL is 4:3.

Answers

From the given figure,
the length of segment SP is 5 - 1 = 4,
the length of segment SR is 8 - 1 = 7,
the length of segment QR is 5 - 1 = 4,
the length of segment PQ is 8 - 1 = 7.

The length of segment MJ is 4 - 1 = 3,
the length of segment ML is 8 - 4 = 4
the length of segment KL is 4 - 1 = 3
the length of segment JK is 8 - 4 = 4

segment SP to segment SR is 4 : 7
segment MJ to segment ML is 3 : 4

segment PQ to segment QR is 7 : 4
segment JK to segment KL is 4 : 3

Because the ratio of any corresponding segments of the two rectangles to another segment is not equal, the two rectangles are not similar.

Therefore, the correct statement is "The two tiles are not similar because segment PQ is to segment QR is 7:4 and segment JK is to segment KL is 4:3"

Identify the range of the function shown in the graph

Answers

Answer:


Step-by-step explanation: the answer is b for apex


On a particular road map 1/2 inch represents 18 miles about how many miles apart are 2 towns that are 21/2 inches apart on this map

Answers

So if 1/2 = 18 miles, we want to first find how many sets of 1/2 are in 21/2. We can just do this by dividing.

(21/2) / (1/2) = 21

So we know that there are 21 sets of 1/2 in 21/2, and since 1/2 and 18 are equal, we can say we know there are 21 sets of 18.

We can multiply to find this.

21 * 18 = 378

So two towns are 378 miles apart.
they are 90miles apart

Choose the biconditional that is equivalent to the definition, "A skyscraper is a very tall building."

Answers

Final answer:

A skyscraper is a very tall building. The biconditional equivalent to this definition is 'A building is a skyscraper if and only if it is very tall.'

Explanation:

The biconditional equivalent to the definition 'A skyscraper is a very tall building' is:

A building is a skyscraper if and only if it is very tall.

This biconditional statement means that for a building to be classified as a skyscraper, it must satisfy the condition of being very tall. Conversely, if a building is very tall, it can be classified as a skyscraper. This definition establishes a two-way relationship between being a skyscraper and being very tall.

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What is m∠JNM?



Enter your answer in the box.

°

Answers

Since they are vertical angles, they are congruent to each other.
Thus, 4x+6=7x-21
Solve for x and get:
x=9
Plug that into the equation that matches the angle and get: 42

we know that

Vertical angles are a pair of opposite and congruent angles formed by intersecting lines

In this problem

m∠JNM=m∠KNL -------> by vertical angles

so

[tex](4x+6)\°=(7x-21)\°[/tex]

Solve for x

[tex]7x-4x=6+21\\3x=27\\x=9\°[/tex]

Find the value of m∠JNM

m∠JNM=[tex](4x+6)\°[/tex]

substitute the value of x

m∠JNM=[tex](4*9+6)\°[/tex]

m∠JNM=[tex]42\°[/tex]

therefore

the answer is

m∠JNM=[tex]42\°[/tex]

The height of a rocket a given number of seconds after it is released is modeled by h (t) = 6t2 + 32t + 10. What does t represent?

Answers

Answer:  t represents the the number of seconds after rocket is released.

Step-by-step explanation:

Given: The height of a rocket a given number of seconds after it is released is modeled by [tex]h (t) = 6t^2 + 32t + 10[/tex].

Here h (height) is the dependent variable , which depends on the number of seconds after rocket is released (independent variable).

Since the independent variable in the function is t, then t must represents the the number of seconds after rocket is released.

The variable t represents the number of seconds that have passed since the rocket was released.

How to identify what a variable represents?

Here we know that the height of a rocket a given number of seconds after it is released is modeled by:

h(t) = 6*t^2 + 32*t + 10

So this is a function that relates height with time in seconds, we know that the function models the height, so we must have that:

[h(t)] is equivalent to height.

This means that the other variable, t, must be related to time in seconds.

Then we can conclude that the variable t represents the number of seconds after the rocket has been released.

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The equation c = 6.5h represents the cost, c, of renting a bicycle for h hours. The table below can be used to show the same information.

If Francesca rents a bicycle for 2 hours and Phil rents a bicycle for 6 hours, how much more does Phil pay?
$13
$26
$39
$52

Answers

c = 6.5h

for 2 hrs...so we sub in 2 for h...for francesca
c = 6.5(2)
c = 13 

for 6 hrs...so we sub in 6 for h...for phil
c = 6.5(6)
c = 39

so Phil pays (39 - 13) = 26 more <=


Answer:

26 (I got 100%)

Step-by-step explanation:

39 - 13

Phil payment - Francesca payment = 26

Parker bought a student discount card for the movies. The card cost $7 and and allows him to buy movie tickets for $5.50. After one month, Parker has spent $89.50 total. How many tickets has he bought since getting the discount card?
A) 12 tickets
B) 14 tickets
C) 15 tickets
D) 16 tickets

Answers

C. is the answer for sure 

A club with 15 women and 12 men need to form a committee that consists of a president, a vice president, a secretary, and a treasurer. how many committees are possible…
a. if the committee must have two women and two men?

Answers

The selection of r objects out of n, can be done in C(n, r) many ways, 

where [tex]\displaystyle{ C(n, r) = \frac{n!}{r!(n-r)!} [/tex], r! being [tex]1\cdot2\cdot...\cdot r[/tex].


Thus, 2 women out of 15 can be selected in a total of C(15, 2) many ways, and 2 men out of 12, can be selected in C(12, 2) many ways.

Any possible pair of women can be combined with any pair of men, thus there are a total of  [tex]C(15, 2)\cdot C(12, 2)[/tex] many ways of forming the committee.

[tex]C(15, 2)\cdot C(12, 2)= \displaystyle{ \frac{15!}{2!13!}\cdot \frac{12!}{2!10!}= \displaystyle{ \frac{15\cdot14\cdot13!}{2!13!}\cdot \frac{12\cdot11\cdot10!}{2!10!}[/tex]
[tex]\displaystyle{ = \frac{15\cdot14}{2}\cdot \frac{12\cdot11}{2}=15\cdot 7\cdot6\cdot11= 6,930[/tex]

Answer: 6,930

What is h(10) equal to?

Answers

h(k) = k^2 - k so we just substitute in 10 for K.

h(10) = 10^2 - 10. 100 - 10 = 90

h(10) = 90. 

A cab company charges an initial rate of $2.50 for a ride, plus $0.40 for each mile driven. What is the equation that models the total fee for using this cab company? Write into an equation and then graph it.

Answers

your equation would be 2.50 + .40x = ?
so you would have 2.50 for the initial rate plus .40 for every mile so that would be represented as .40x . so lets say you took a cab ride 20 miles away so you would have 2.50 +.40(20)=10.50 so your total fee would end up being $10.50

Answer:

The equation that models the total fee for using this cab company

[tex]y=\$2.50+\$0.40\times x[/tex]

Step-by-step explanation:

Initial rate charged by company = $2.50

Amount charged for an each mile = $0.40

Let the miles cover during a ride = x

Total cost of ride can given as = y

The equation that models the total fee for using this cab company

[tex]y=\$2.50+\$0.40\times x[/tex]

The graphical interpretation equation in an image.

A line has a slope of 5 and a y-intercept of 4 what is the equation in slope intercept form

Write your answer using integers proper fraction and improper fractions in simplest form

Answers

given slope(m) is 5 and y-intercept(b) is 4. so you simply plug in where it belongs into the equation, y = mx + b

y = 5x + 4

hope this helped, God bless!

Write the equation in the slope-intercept form. 7x − 4y + 8 = 0

Answers

Final answer:

To convert the equation 7x - 4y + 8 = 0 to slope-intercept form, solve for y to get y = (7/4)x + 2, with a slope of 7/4 and a y-intercept of 2.

Explanation:

To write the equation 7x − 4y + 8 = 0 in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we want to solve for y. The steps are as follows:

Subtract 7x and 8 from both sides of the equation to isolate terms involving y: -4y = -7x - 8.Divide every term by -4 to solve for y: y = (7/4)x + 2.

Thus, the equation of the line in slope-intercept form is y = (7/4)x + 2. Here, the slope is 7/4 and the y-intercept is 2.

PLEASE HELP ASAP: A particle is moving with velocity v(t) = t2 – 9t + 18 with distance, s measured in meters, left or right of zero, and t measured in seconds, with t between 0 and 8 seconds inclusive. The position at time t = 0 sec is 1 meter right of zero, that is, s(0) = 1.

The average velocity over the interval 0 to 8 seconds

The instantaneous velocity and speed at time 5 secs

The time interval(s) when the particle is moving right

The time interval(s) when the particle is
going faster
slowing down

Find the total distance the particle has traveled between 0 and 8 seconds

Answers

Answer:

1) Average velocity = 10/3 m/s

2) Instantaneous velocity = -2 m/s
   Speed = 2 m/s to the left

3) (0, 3) ∪ (6, 8]

4) Going faster: (3, 4.5) ∪ (6, 8]
   Slowing down: (0, 3) ∪ (4.5, 6)

5) Total distance = 35.67 m (nearest hundredth)

Step-by-step explanation:

The relationships between position (displacement), velocity and acceleration are:

[tex]\boxed{\boxed{\begin{array}{c}\textbf{POSITION (s)}\\\\\text{Differentiate} \downarrow\qquad\uparrow\text{Integrate}\\\\\textbf{VELOCITY (v)}\\\\\text{Differentiate}\downarrow\qquad\uparrow \text{Integrate}\\\\\textbf{ACCELERATION (a)}\end{array}}}[/tex]

Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its position s(t) we can integrate v(t):

[tex]\begin{aligned}\displaystyle s(t)=\int v(t)\;\text{d}t&=\int(t^2-9t+18)\;\text{d}t\\\\&=\dfrac{t^{2+1}}{2+1}-\dfrac{9t^{1+1}}{1+1}+18t+C\\\\&=\dfrac{t^{3}}{3}-\dfrac{9t^{2}}{2}+18t+C\end{aligned}[/tex]

As s(0) = 1, then:

[tex]\begin{aligned}s(0)=\dfrac{(0)^{3}}{3}-\dfrac{9(0)^{2}}{2}+18(0)+C&=1\\0-0+0+C&=1\\C&=1\end{aligned}[/tex]

Therefore, the position function s(t) is:

[tex]\large\boxed{s(t)=\dfrac{t^3}{3}-\dfrac{9t^2}{2}+18t+1}[/tex]

Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its acceleration a(t) we can differentiate v(t):

[tex]\begin{aligned}a(t)=\dfrac{\text{d}}{\text{d}t}[v(t)]&=2\cdot t^{2-1}-1\cdot9t^{1-1}+0\\&=2t-9\end{aligned}[/tex]

Therefore, the acceleration function a(t) is:

[tex]\large\boxed{a(t)=2t-9}[/tex]

[tex]\hrulefill[/tex]

Question 1

To find the average velocity over the interval [0, 8], use the formula:

[tex]\textsf{Average Velocity}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}[/tex]

In this case:

t₁ = 0t₂ = 8

Calculate the position at t₁ and t₂ by substituting t = 0 and t = 8 into s(t):

[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1}=1[/tex]

[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1}=\dfrac{83}{3}[/tex]

Therefore:

[tex]\textsf{Average Velocity}=\dfrac{s(8)-s(0)}{8-0}=\dfrac{\frac{83}{3}-1}{8}=\dfrac{10}{3}\; \sf m/s[/tex]

Therefore, the average velocity is 10/3 m/s.

[tex]\hrulefill[/tex]

Question 2

To find the instantaneous velocity at t = 5 seconds, substitute t = 5 into v(t):

[tex]\begin{aligned}v(5)&=(5)^2-9(5)+18\\&=25-45+18\\&=-2\end{aligned}[/tex]

So, the instantaneous velocity at t = 5 seconds is -2 m/s.

Speed is a scalar quantity that measures how fast an object is moving regardless of its direction. Therefore, speed is the magnitude of velocity:

[tex]\textsf{Speed}=|v(5)|=|-2|=2\;\sf m/s[/tex]

Therefore, the speed at t = 5 is 2 m/s to the left.

[tex]\hrulefill[/tex]

Question 2

The particle changes direction when v(t) = 0.

[tex]\begin{aligned}v(t)&=0\\\implies t^2-9t+18&=0\\t^2-6t-3t+18&=0\\t(t-6)-3(t-6)&=0\\(t-3)(t-6)&=0\\\\t-3&=0\implies t=3\\t-6&=0\implies t=6\end{aligned}[/tex]

Therefore, the particle changes direction at t = 3 and t = 6.

We know that the position of the particle at t = 0 is 1 meter right of zero. Therefore:

It is moving to the right in the interval (0, 3).It is moving to the left in the interval (3, 6).It is moving to the right in the interval (6, 8].

Therefore, the time intervals between 0 ≤ t ≤ 8 when the particle is moving right is:

(0, 3) ∪ (6, 8]

[tex]\hrulefill[/tex]

Question 4

When a(t) > 0:

[tex]\begin{aligned}a(t)& > 0\\2t-9& > 0\\2t& > 9\\t& > \dfrac{9}{2}\\t& > 4.5\; \sf s\end{aligned}[/tex]

When a(t) < 0:

[tex]\begin{aligned}a(t)& < 0\\2t-9& < 0\\2t& < 9\\t& < \dfrac{9}{2}\\t& < 4.5\; \sf s\end{aligned}[/tex]

Therefore:

Velocity is positive in the interval (0, 3) and (6, 8].Velocity is negative in the interval (3, 6).Acceleration is positive in the interval (4.5, 8].Acceleration is negative in the interval (0, 4.5).

(Refer to the attachment).

If velocity and acceleration have the same sign, it means the object is speeding up.

If velocity and acceleration have opposite signs, it means the object is slowing down.

Therefore, the time intervals when the particle is going faster and slowing down are:

Going faster: (3, 4.5) ∪ (6, 8]Slowing down: (0, 3) ∪ (4.5, 6)

[tex]\hrulefill[/tex]

Question 5

To find the total distance the particle has traveled between 0 and 8 seconds, we need to consider the distance traveled between the intervals when it changes direction.

To do this, find the position of the particle at t = 0, t = 3, t = 6 and t = 8.

[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1=1[/tex]

[tex]s(3)=\dfrac{(3)^3}{3}-\dfrac{9(3)^2}{2}+18(3)+1=23.5[/tex]

[tex]s(6)=\dfrac{(6)^3}{3}-\dfrac{9(6)^2}{2}+18(6)+1=19[/tex]

[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1=\dfrac{83}{3}\approx27.67[/tex]

Therefore, in the interval 0 ≤ t < 3, the particle travels:

[tex]|s(3)-s(0)|=|23.5-1|=22.5\; \sf meters\;(to\;the\;right)[/tex]

In the interval  3 < t < 6, it travels:

[tex]|s(6)-s(3)|=|19-23.5|=4.5\; \sf meters\;(to\;the\;left)[/tex]

In the interval 6 < t ≤ 8, it travels:

[tex]|s(8)-s(6)|=|27.67-19|=8.67\; \sf meters\;(to\;the\;right)[/tex]

So the total distance the particle has traveled between 0 and 8 seconds is:

[tex]\textsf{Total distance}=22.5+4.5+8.67=35.67\; \sf meters[/tex]

You have 6 pints of glaze. It takes 7/8 of a pint to glaze a bowl and 9/16 of a pint to glaze a plate. You want to glaze 5 bowls, and then use the rest for plates. How many plates can you glaze? How much glaze will be left over?

Answers

The answer is 2 plates and .50 glaze left over.

How did I get this?

First lets calculate how many bowls can we glaze. 
5 bowls * 7/8 of a pint = 4.375 pints
4.375 = how many glaze we will need to make 5 bowls.

6 pints - 4.375pints = 1.625 pints left over after we glaze the bowls. 

Now, we need to figure out how many plates we can make with 1.625 glaze.

P X 9/16 = 1.625
Multiply both sides by 16/9 to isolate the variable.
P = 1.625*16/9 
P = 2.88 plates

So, we can only glaze 2 plates. 

How many glaze is left over?
First, how many glaze did we have left over from the bowls? 1.625 glaze
Then, how many plates did we make? 2 at .5625 each = 1.125 glaze

1.625 - 1.125 = .50 glaze left over.

The answer is 2 plates and .50 glaze left over.

2 plates can be glazed and 0.50 glaze will be left over.

What is Unitary Method?

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

for example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Total bowls= 5

fraction of pint to glaze bowl= 7/8

So, pint to glaze 5 bowls

= 5 bowls x  7/8 of a pint

=  4.375 pints

and, left over after glazing the bowls.

= 6 pints - 4.375pints

= 1.625

Now, we know to glaze plate pint used= 9/16

So, number of plates will be  = 1.625 x 16/9

                                                 = 2.88 plates

Thus, we can only glaze 2 plates.

Also,  glaze left over

= 1.625 - 1.125

= 0.50 glaze left over.

Learn more about unitary method here:

https://brainly.com/question/28276953

#SPJ5

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