Find the sum of the whole numbers from 1 to 960

[tex]g(x) = 2(x-2)^2-5 \text{ is equivalent to } 2x^2-8x+3[/tex]

Solution:

Given expression is:

[tex]f(x) = 2x^2-8x+3[/tex]

We have to find the equivalent expression

[tex]\mathrm{Write}\:2x^2-8x+3\:\mathrm{in\:the\:form:\:\:}x^2+2ax+a^2[/tex]

[tex]f(x) = 2x^2-8x+3[/tex]

[tex]\mathrm{Factor\:out\:}2[/tex]

[tex]2\left(x^2-4x+\frac{3}{2}\right)[/tex]

[tex]\mathrm{Add\:and\:subtract}\:\left(-2\right)^2\:[/tex]

[tex]2\left(x^2-4x+\frac{3}{2}+\left(-2\right)^2-\left(-2\right)^2\right)[/tex]

Simplify the above equation

[tex]2(x^2-4x + \frac{3}{2} + 4 -4)\\\\2(x^2-4x +4 + \frac{3}{2} - 4)\\\\2(x^2-4x +4 -\frac{5}{2}) ------- eqn 1[/tex]

Using the algebraic identity,

[tex]x^2+2ax+a^2=\left(x+a\right)^2[/tex]

Therefore,

[tex]x^2-4x+4=\left(x-2\right)^2[/tex]

Substitute the above in eqn 1

[tex]2((x-2)^2-\frac{5}{2})[/tex]

Simplify the above equation by multiplying 2 with terms inside the bracket

[tex]2(x-2)^2 -5[/tex]

Thus the equivalent expression is [tex]g(x) = 2(x-2)^2-5[/tex]

The solution is x = 7 and y= 3

Solution:

Given pairs of equation are:

x = 1 + 2y ------- eqn 1

2x + y = 17 ------- eqn 2

We have to use algebraic method

Algebraic method involves substitution and elimination method

Let us use the substitution method

From equation 1, we can see that x is equal to 1 + 2y. If we substitute this for y in equation 2, we have:

2(1 + 2y) + y = 17

Solve for brackets

2 + 4y + y = 17

Combine the like terms which has same variable

2 + 5y = 17

5y = 17 - 2

5y = 15

Divide both sides of equation by 5

y = 3

So the value of y is 3. This value of y can be substituted into either equation 1 or 2 to find the value of x

Substitute y = 3 in eqn 1

x = 1 + 2(3)

x = 1 + 6

x = 7

So the solution is x = 7 and y = 3

Answer:

Step-by-step explanation:

Here, an equation of straight line is given.

The equation is [tex]y = -\frac{3x}{2}[/tex].

If you put x = 0, in the above equation, you will get y = 0.

Hence, the equation passes through (0, 0).

Again if we put, x = 1 in the above equation, we will get y = - 1.5.

It means the equation also passes through (1, -1.5).

Joining these two points, you can graph the straight line.

Answer:

28.56

Step-by-step explanation:

Find the perimeter of the square:

4s

= 4×4

=16

Find the circumference of the circle and since it's half circle for both side it's going to be 1 circle:

3.14×d

= 3.14×4

=12.56

Add perimeter of square and circumference:

16+12.56

=28.56

It might help to think of this shape as a track for runners to run around, even though the scale is much too small for it.

The two semicircles on each end can be combined to form a full circle. This circle has radius r = 2, which is half of the diameter 4 (square's side length).

C = 2*pi*r

C = 2*pi*2

C = 4pi is the exact circumference

C = 4*3.14

C = 12.56 is the approximate circumference

This represents the distance around the curved portions of the track.

The straight portions are 4 each. We do not include the vertical sides as they are not on the outside, and the runners do not run along those vertical portions (they run around the curved parts instead). So we add on 4+4 = 8 to the previous value 12.56 to get 20.56 as our final answer.

Answer:

The length of the rectangle is 76 feet and the width of the rectangle was 25 feet that is extended by x feet.

Step-by-step explanation:

The total area, in square feet, of a rectangular stage that has been widened by x feet, is represented by 1,900 + 76x.

Now, factorizing the expression for the area we get,

A = 1900 + 76x

⇒ A = 76(25 + x)

Therefore, from the above expression, we can assume that the length of the rectangle is 76 feet and the width of the rectangle was 25 feet that is extended by x feet. (Answer)

45/60 and 10/60 because we know that the common denominator is 60 then 3/4 of an hour is 45mins then 1/6 of an hour is 10

By finding common denominators, we can express the boat drawbridge opening times in terms of a shared timeframe, such as every 60 minutes or 30 minutes. The common denominator is the smallest number that the opening times can be equally divided into.

The question relates to the application of common denominators in the context of timekeeping. Without specific times for the opening of Colby and Wave drawbridge, I will give a hypothetical scenario as an example.

Let's say Colby drawbridge opens every 20 minutes and Wave drawbridge opens every 15 minutes. A common denominator of 20 and 15 is 60, as they both divide evenly into 60. So, you could rename the opening times in terms of 60 minutes: Colby drawbridge opens 3 times every 60 minutes (60/20) and Wave drawbridge opens 4 times every 60 minutes (60/15).

If we want to use another common denominator, you could use 30. Then Colby drawbridge opens 1.5 times every 30 minutes (30/20) and Wave drawbridge opens 2 times every 30 minutes (30/15).

The way to find common denominators involves looking for the smallest number that both of the numbers can divide equally into. In this case, 60 was an obvious choice since there are 60 minutes in an hour, but other common denominators like 30 can also be used.

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Rachel needs 1776 inches of board

Solution:

Given that, Rachel is making a planter box

Her box measures 345 inches long and 543 inches wide

Length = 345 inches

Width = 543 inches

To find: Inches of board she needs

So we need to find the perimeter of the board

The board is usually of rectangular shape

Perimeter of rectangle is given as:

Perimeter = 2(length + width)

Perimeter = 2(345 + 543)

Perimeter = 2(888)

Perimeter = 1776

Thus she needs 1776 inches of board

Answer:

92 ft squared

Step-by-step explanation:

6*7=42

5*10=50

42+50=92 ft squared

Answer:

120

Step-by-step explanation:

hope this helps :) have a good day

Answer:

4/6

Step-by-step explanation: numerator is the top number and denominator is the bottom number so in this case 4 is the numerator and 6 is the denominator.

The fraction with a numerator of 4 and a denominator of 6 is 4/6, which simplifies to 2/3.

To write a fraction with 4 as the numerator and 6 as the denominator, we simply place 4 above 6, resulting in the fraction 4/6. This fraction can be simplified by dividing both the numerator and denominator by their greatest common factor, which in this case is 2. Therefore, the simplified fraction is 2/3.

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The linear equation that models the value of the automobile in terms of the year x is y = -1,110 x + 14,220

Step-by-step explanation:

The average value of a certain type of automobile was 14,220 in 1993 and depreciated to 9,780 in 1997

We need to write and graph a linear equation that models the value of the automobile in terms of the year x

The form of the linear equation is y = mx + b, where b is the initial value (value y at x = 0), and m is the rate of change

∵ The average value the of automobile was 14,220 in 1993

∵ x = 0 represents 1993

∴ The point which represent the data is (0 , 14,220)

- b is the value of y at x = 0

∴ b = 14,220

∵ The average automobile was 9,780 in 1997

- To find x subtract 1993 from 1997

∵ 1997 - 1993 = 4

∴ x = 4

∴ The point which represent the data is (4 , 9,780)

∵ m = Δy/Δx

∴ [tex]m=\frac{9780-14220}{4-0}[/tex]

∴ m = -1,110

- Substitute the values of m and b in the form of the equation

∴ y = -1,110 x + 14,220

The linear equation that models the value of the automobile in terms of the year x is y = -1,110 x + 14,220

The graph is attached below

Each square unit represents 1000 in the graph

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A linear function is a function that changes at a constant rate.

The equation of the linear function is [tex]y = -1110x + 14220[/tex]

The given parameters are:

Start by calculating the slope (m)

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

This gives

[tex]m = \frac{9780 - 14220}{4- 0}[/tex]

Simplify

[tex]m = \frac{-4440}{4}[/tex]

This gives

[tex]m = -1110[/tex]

The equation is then calculated as:

[tex]y = m(x -x_1) + y_1[/tex]

This gives

[tex]y = -1110(x -0) + 14220[/tex]

Expand

[tex]y = -1110x + 14220[/tex]

Hence, the equation of the linear relationship is [tex]y = -1110x + 14220[/tex]

See attachment for the graph

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Answer:

Therefore, 3 feet less than the Legnth is

[tex]Length - 3[/tex]  OR

[tex]L - 3[/tex]

Step-by-step explanation:

What is 3 feet less than the Legnth  ?

Solution:

It is the Algebraic Expression in Statement form.

Algebraic Expression is given in the form numbers and variables with the operands such as,

' + ' Plus Sign

' - ' Minus Sign

' × ' Multiplication Sign

' ÷ ' Division Sign, etc.

Here let Length be the variable denoted as 'L'

So .3 feet less means Subtract 3,

'Than the Length' means from the Length,

Then the Expression will be

[tex]Length - 3[/tex]  OR

[tex]L - 3[/tex]

Therefore, 3 feet less than the Legnth is

[tex]Length - 3[/tex]  OR

[tex]L - 3[/tex]

A general term used to refer to buying and selling of homes is the house market.

Final answer:

The term 'real estate' is generally used to describe the buying and selling of homes, covering both new home construction, which impacts real GDP, and the resale of existing homes, which is a significant component of the housing market.

Explanation:

The general term commonly used to refer to the buying and selling of homes is real estate. This term encompasses the market for new homes, which are part of real gross domestic product (real GDP), as well as the resale of existing homes. While the buying and selling of existing homes is not counted in real GDP, it is a significant part of the housing market. Real estate can be influenced by factors such as supply and demand, economic conditions, and demographics. The American Association of Realtors suggests that when there is a six months' supply of houses, the market is considered to be in equilibrium, which can help determine whether housing prices are likely to rise or fall.

Answer:

$681

Step-by-step explanation:

619.100 * 10% = 61.91

619.100 + 61.91 = 681.01

Rounding it down because it is not over 5. $681

Therefore your answer will be $681

Hope this helps!

Answer: There were 22 cats and 3 dogs at the shelter on Tuesday.

Step-by-step explanation:

Let be "c" the number of cats at the shelter on Tuesday and  "d" the  number of dogs at the shelter on Tuesday.

Set up a System of equations:

[tex]\left \{ {{c+d=25} \atop {4.5c+8d=123}} \right.[/tex]

You can use the Elimination Method to solve the System of equations.

The steps are:

1. Multiply the first equation by -8.

2. Add the equations.

3. Solve for the variable "c".

Then:

[tex]\left \{ {{-8c-8d=-200} \atop {4.5c+8d=123}} \right.\\....................\\-3.5c=-23\\\\c=22[/tex]

4. Substitute the value of "c" into the first original equation.

5. Solve for the variable "d" in order to find its value.

Then, you get:

[tex]22+d=25\\\\d=25-22\\\\d=3[/tex]

Answer:

This is not a direct variation

Step-by-step explanation:

if 12 dollars for 6 bagels is true then each bagel is $2

if 9 dollars for 24 bagels then each bagel is $0.38

Step-by-step explanation:

[tex]|a|=\left\{\begin{array}{ccc}a&if&a\geq0\\-a&if&a<0\end{array}\right\\\\x\geq0,\ \text{then}\ |x|=x\\\\x+|x|=x+x=2x\geq0[/tex]

Answer

Josh's textbook reached the ground first

Josh's textbook reached the ground first by a difference of [tex]t=0.6482[/tex]

Step-by-step explanation:

Before we proceed let us re write correctly the height equation which in correct form reads:

[tex]h(t)=-16t^2 +v_{o}t+s[/tex]       Eqn(1).

Where:

[tex]h(t)[/tex] : is the height range as a function of time

[tex]v_{o}[/tex]   : is the initial velocity

[tex]s[/tex]     : is the initial heightin feet and is given as 40 feet, thus Eqn(1). becomes:

[tex]h(t)=-16t^2 + v_{o}t + 40[/tex]        Eqn(2).

Now let us use the given information and set up our equations for Ben and Josh.

Ben:

We know that [tex]v_{o}=60ft/s[/tex]

Thus Eqn. (2) becomes:

[tex]h(t)=-16t^2+60t+40[/tex]        Eqn.(3)

Josh:

We know that [tex]v_{o}=48ft/s[/tex]

Thus Eqn. (2) becomes:

[tex]h(t)=-16t^2+48t+40[/tex]       Eqn. (4).

Now since we want to find whose textbook reaches the ground first and by how many seconds we need to solve each equation (i.e. Eqns. (3) and (4)) at [tex]h(t)=0[/tex]. Now since both are quadratic equations we will solve one showing the full method which can be repeated for the other one.

Thus we have for Ben, Eqn. (3) gives:

[tex]h(t)=0<=>-16t^2+60t+40=0[/tex]

Using the quadratic expression to find the roots of the quadratic we have:

[tex]t_{1,2}=\frac{-b+/-\sqrt{b^2-4ac} }{2a} \\t_{1,2}=\frac{-60+/-\sqrt{60^2-4(-16)(40)} }{2(-16)} \\t_{1,2}=\frac{-60+/-\sqrt{6160} }{-32} \\t_{1,2}=\frac{15+/-\sqrt{385} }{8}\\\\t_{1}=4.3276 sec\\t_{2}=-0.5776 sec[/tex]

Since time can only be positive we reject the [tex]t_{2}[/tex] solution and we keep that Ben's book took [tex]t=4.3276[/tex] seconds to reach the ground.

Similarly solving for Josh we obtain

[tex]t_{1}=3.6794sec\\t_{2}=-0.6794sec[/tex]

Thus again we reject the negative and keep the positive solution, so Josh's book took [tex]t=3.6794[/tex] seconds to reach the ground.

Then we can find the difference between Ben and Josh times as

[tex]t_{Ben}-t_{Josh}= 4.3276 - 3.6794 = 0.6482[/tex]

So to answer the original question:

Whose textbook reaches the ground first and by how many seconds?

Answer:

The pharmacy should give 3.0 mL solution to the patient

Step-by-step explanation:

Given,

750 mg of penicillin in 5 mL solutions

Hence, amount of penicillin in 1 mL of solution= [tex]\frac{750}{5}[/tex] = 150 mg/ mL

So, amount of solution for 450 mg of penicillin= [tex]\frac{450}{150}[/tex] = 3.0 mL

Answer:

B. [tex](x,y)\rightarrow (x-4,y+3)[/tex]

Step-by-step explanation:

To get from point A to point A', you need to move 4 unite to the left and 3 units up.

To get from point B to point B', you need to move 4 unite to the left and 3 units up.

To get from point C to point C', you need to move 4 unite to the left and 3 units up.

To get from point D to point D', you need to move 4 unite to the left and 3 units up.

Hence, the translation rule is

[tex](x,y)\rightarrow (x-4,y+3)[/tex]

Answer:

12 times k equals 84

Hope this helps

-Amelia

Answer:

Although our example says the correct answer is 2/3 x 1/2, remember, with multiplying ... If we divide both 525 and 700 by 175, we can see that 525/700 is equal to 3/4.

Step-by-step explanation:

Final answer:

To solve '4/6 times what equals 1/6', you need to find the reciprocal of 4/6 and multiply it by 1/6. The reciprocal of 4/6 is 6/4, and when you multiply this by 1/6, you get 1/4. Thus, 4/6 times 1/4 equals 1/6.

Explanation:

The question asks for the unknown value that when multiplied by 4/6 gives the result of 1/6. To find the unknown value, you would set up the equation:

[tex]\(\frac{4}{6} \times x = \frac{1}{6}\)[/tex]

To find the value of x, you can divide both sides of the equation by 4/6. In other words, you are looking for the reciprocal of 4/6 that, when multiplied by 1/6, gives 1. The reciprocal of 4/6 is 6/4. Thus, when you perform the division, you get:

[tex]\(x = \frac{1}{6} \times \frac{6}{4}\)[/tex]

To simplify, you multiply the numerators and the denominators:

[tex]\(x = \frac{1 \times 6}{6 \times 4}\)[/tex]

[tex]\(x = \frac{6}{24}\)[/tex]

Since 6 and 24 have a common factor of 6, you can simplify the fraction:

[tex]\(x = \frac{1}{4}\)[/tex]

Therefore, 4/6 times 1/4 equals 1/6.

Answer:

A - 2B

Step-by-step explanation:

3x + 2y = A

5x + y = B

The first equation has 2y. The second equation has y.

If you multiply the second equation by 2, it will have 2y.

Then you subtract teh second equation from the first equation.

2y - 2y = 0, so you eliminate the y variable.

********

3x + 2y = A

2(5x + y) = 2B

********

3x + 2y = A

10x + 2y = 2B

********

Subtract the second equation from the first equation.

3x + 2y - (10x + 2y) = A - 2B

3x - 10x + 2y - 2y = A - 2B

-7x = A - 2B

Answer: A - 2B

Final answer:

To determine Zoe's share from the 3:7 tip ratio when Hannah received £24, divide £24 by 7 to find the value of one part of the ratio, then multiply by 3 to find Zoe's share, resulting in £10.29.

Explanation:

When Zoe and Hannah share tips at a ratio of 3:7 and it is known that Hannah received £24, we can set up a proportion to find out how much Zoe received. Since Hannah's share of the tips corresponds to the 7 parts of the ratio, each part is worth £24 divided by 7 (which is Hannah's portion of the ratio). To find out Zoe's share, we multiply the value of each part by 3 (which is Zoe's portion of the ratio).

Calculate the value of one part of the ratio: £24 / 7 = £3.43 (rounded to two decimal places).

Calculate Zoe's share: £3.43 * 3 = £10.29 (rounded to two decimal places).

Therefore, Zoe's share of the tips would be £10.29.

Answer:

From Walter's steps we have that Step2: Reason 2 is incorrect.

The correct reason is Subtraction property of equality (we are using this property only not addition property of equality )

Step-by-step explanation:

Given equation is three halves times x plus ten equals forty

It can be written as below

[tex]\frac{3}{2}x+10=40[/tex]

To solve the given equation :

Step1 Reason1 : [tex]\frac{3}{2}x+10=40[/tex]

Step2 Reason2: Subtracting 10 on both sides of the above equation we get

[tex]\frac{3}{2}x+10-10=40-10[/tex] which is Subtraction property of equality

Step3 Reason3: [tex]\frac{3}{2}x=30[/tex]

Step4 Reason4: Multiply the above equation into [tex]\frac{2}{3}[/tex] on both sides we get

[tex]\frac{3}{2}x\times \frac{2}{3}=30\times \frac{2}{3}[/tex]

which is the Multiplication Property of Equality

Step5 Reason5: Simplify the above equation we get

 [tex]x= \frac{60}{3}[/tex]

Therefore x=20

From Walter's steps we have that Step2: Reason 2 is incorrect.

The correct reason is Subtraction property of equality (we are using this property only not addition property of equality )

Answer:

b

Step-by-step explanation:

Answer:

Yes it can.

Step-by-step explanation:

We plot the data given on the graph and see whether there is any correlation.

The data is graphed in the figure attached. Looking at the graph we see that the points do seem to lie on a straight line which I have drawn as a dashed line.  

The dashed line represents the function

[tex]f(x)=-2x+40[/tex]

The function tells us that as Khaled gives more homework to his students, less students complete their homework.

The store bought 90 half gallons of milk.

To find the number of half gallons of milk the store bought, we can set up an equation using the given information.

Let x be the total number of half gallons of milk bought by the store.

The store bought each half gallon of milk for $2.80, so the total cost of x half gallons of milk is 2.80x dollars.

After the refrigerator malfunction, 12 half gallons of milk were ruined. So the remaining milk is x - 12 half gallons.

The remaining milk is sold for $4.00 per half gallon, so the total revenue from selling the remaining milk is 4(x - 12) dollars.

The profit made by the store is given as $60.00, so the equation becomes:

4(x - 12) - 2.80x = 60

Simplifying the equation: 4x - 48 - 2.80x = 60

Combining like terms: 1.20x - 48 = 60

Adding 48 to both sides: 1.20x = 108

Dividing both sides by 1.20: x = 90

Therefore, the store bought 90 half gallons of milk.

Answer:

c 2x+1

Step-by-step explanation:

Answer:

The dimensions of the original sheet of paper are 8 inches by 6 inches.

The total area of the sheet of paper = 8 inches [tex]\times[/tex] 6 inches  = 48 [tex]inches^{2}[/tex]

Step-by-step explanation:

i) the two parts of the sheet of paper are 6 inches by 5 inches and 3 inches by   6 inches.

ii) therefore we see that the both parts have a 6 inch side.

iii) therefore we can say that the width of the sheet of paper is = 6 inches

iv) therefore we can also say that the length of the sheet of paper =  3  +  5 inches  = 8 inches.

v) therefore the dimensions of the original sheet of paper are 8 inches by 6 inches.

vi) therefore the total area of the sheet of paper = 8 inches [tex]\times[/tex] 6 inches

                                                                          = 48 [tex]inches^{2}[/tex]

Answer:

6. The equation needs to be correct for every pair (x, y) of values from the table. If we want to find the right equation, we simply plug a pair from the table and see if it fits. Let's take the first pair (1, -2) and plug it in the equation a):

-2 = 1+2 ---> obviously not correct

Now let's do the same for equation b):

-2 = 3+3 ---> again incorrect

Equation c):

-2 =1-11 --> incorrect

And equation d):

-2 = 2-4 ---> correct

It's important to know that we could've chosen any pair from the table to check the equation. However, sometimes one pair can fit into more than one equation. In that case, we have to plug in other pairs as well to see which equation is the right one.

7. Scale factor, simply put, shows how many times one value of the figure changed when mapping onto another.  

We are given the triangle ABC. Let's say the side of each of these little squares in 1 unit. That way, we conclude that the length AB is 4 units and BC is 3 units.

Now, let's look at the triangle A'B'C'. Length of the A'B' is 12 units and B'C' is 9 units.

Now to find the scale factor we simply have to find how many times are the sides of the mapped triangle greater then the triangle ABC:

AB/A'B' = BC/B'C' = 3

So, the scale factor is 3.

8. Note that the plumber charges two fees:

- base fee is a constant fee, for all appointments, regardless of the repair needed and hours spent.

- repair fee is a fee paid only if the repair is needed and is dependent on the number of hours spent repairing (more hours greater the fee)

So, in our equation

y = 25x + 30

y represents the total cost and x represents hours.

Now, we can see that this 25x part of equation represents repair fee, because it depends on x (for every hour spent, one pays another $25).

That means that 30 from our equation represents the base fee, the amount everyone will be charged regardless of the need for repair.

9. A relationship is linear if it's graph is a straight line, which means that its rate of change is constant. That means that the change in values of x of the two adjacent points, divided by the change of y is always the same number. That means that:

(x2 - x1)/(y2-y1) = (x3 - x2)/(y3 - y2) = (x4 - x3)/(y4 - y3) etc.

Let's test this.

Our first point is (0, 1) and next one is 1, 2). So, our first change of rate is:

(1-0)/(2-1) = 1/1 = 1

Let's find the rate of change for the next two adjacent pairs:

(x2, y2) = (1, 2)

(x3, y3) = (2, 4)

So, the rate of change is:

(2-1)/(4-2) = 1/2  

It's obvious that the rate of change isn't always the same, which means this relation isn't linear.

36 chairs are left

Solution:

Given that, 240 beach chairs are on sale

85% of the chairs are sold

To find: remaining chairs

Total number of chairs = 240

Number of chairs sold = 85 % of total chairs

Number of chairs sold = 85 % of 240

[tex]\rightarrow 85 \% \text{ of } 240\\\\\rightarrow \frac{85}{100} \times 240\\\\\rightarrow 204[/tex]

Thus 204 chairs are sold

Remaining chiars = total chairs - sold chairs

Remaining chairs = 240 - 204 = 36

Thus 36 chairs are left

36 chairs are left

the store sold 204 chairs