Two parts: Make sure to answer EACH part:

1. What is the rate of change for each shop shown below?

2. Which shop has the GREATEST rate of change?

Answer:

2   6    12   20,000

6   18   36   60,000

Step-by-step explanation:

The bottom number is 3 times the top number.

The standard form of the quadratic function y=ax² + bx + c = a (x - h)² + k

By definition, quadratic function is a function in the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero.

ax² + bx + c = a (x² - 2xh + h²) + kax² + bx + c

= ax² - 2ah x + (ah² + k)

Comparing the coefficients of x on both sides,b = -2ah ⇒ h = -b/2a .............................................(i)

Comparing the constants on both sides,c = ah² + kc = a (-b/2a)² + k (From (1))

c = b²/(4a) + kk = c - (b²/4a)k = (4ac - b²) / (4a)

Therefore, we can use the formulas h = -b/2a and k = (4ac - b²) / (4a) to convert the standard form to its vertex form.

the vertex form of the parabola is  y = a (x - h)² + k

The standard form of the parabola is y = ax² + bx + c

The standard form of the quadratic function y=ax²+bx+c is x=-b± [tex]\frac{\sqrt{b^{2}- } 4ac}{2a}[/tex]

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Answer:

D, (5,-8)

Step-by-step explanation:

On the graph in the image, you can see option D's only point on the line.

Answer: D

Step-by-step explanation:

You will plug every value into the equation; if they are true, that is the correct option.

A.

5+8=3/7(3/7-5)

13=3/7(-32/7)

13=(-96/49)

A is INCORRECT

B.

8+8=3/7(-5-5)

16=3/7(-10)

16=-30/7

B is INCORRECT

C.

-5+8=3/7(8-5)

3=3/7(3)

3=9/7

C is INCORRECT

D.

-8+8=3/7(5-5)

0=3/7(0)

0=0

D IS CORRECT

The volume of the largest box that can be made from the rectangular cardboard obtained by using the value of the derivative of the function for the volume at the extremum point is 18 cubic feet

An extremum of a function is the point where the value of the function is a maximum or a minimum.

The side length of the cut out squares = x

The width of the rectangular cardboard = 5 feet

The length of the rectangular cardboard = 8 feet  

The length of the side of the cut square = 8 - 2·x

Width of the side of the cut square = 5 - 2·x

Height of the cut square = x

Volume of the cut square, V = (8 - 2·x) × (5 - 2·x) × x = 4·x³ - 26·x² + 40·x

When the volume of the box is largest, we have;[tex]V' = \dfrac{d}{dx} \left(4\cdot x^3 - 26\cdot x^2 + 40\cdot x\right) = 12\cdot x^2 - 52\cdot x + 40=0[/tex]

12·x² - 52·x + 40 = 0

3·x² - 13·x + 10 = 0

Which gives;

(3·x - 10)·(x - 1) = 0

The dimensions of the cut square that gives the maximum volume are therefore;

x = 1 or x = 10/3

When x = 10/3, we have;

V = (8 - 2×(10/3)) × (5 - 2×(10/3)) × (10/3) ≈ -200/27 < 0

When x = 1, we have;

V = (8 - 2×1) × (5 - 2×1) × 1 = 18

The value of x that gives the maximum volume is therefore, x = 1

The dimensions of the box that give the maximum volume are therefore;

Height = x = 1

Length = 8 - 2×1 = 6

Width = 5 - 2 × 1 = 3

The dimensions of the box that gives the maximum volume are;

Height = 1 ft

Length = 6 ft

Width = 3 ft

The volume of the largest box that can be formed is 18 cubic feet

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The linear function that models the puppy's growth is written as: y = 3x + 3.

The values we need to find before we can write he linear function that models a situation, are:

The linear function would be expressed as y = mx + b, which is in the slope-intercept form.

The table of the puppy's growth that Shayla is tracking is shown below. Using two points on the table, say, (0, 3) and (1, 6):

Slope (m) = change in y / change in x = (6 - 3)/(1 - 0)

Slope (m) = 3/1 = 3

Substitute m = 3 and (0, 3) into y = mx + b to find b:

3 = 3(0) + b

3 = 0 + b

3 = b

b = 3

To write the linear function, substitute m = 3 and b = 3 into the equation, y = mx + b:

y = 3x + 3

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Answer: A

Step-by-step explanation: FLVS question the answer is A

The corresponding values of F1 and F2 in the ordered pairs are 77 degrees Fahrenheit and -22 degrees Fahrenheit

Given: the equation F=9/5C+32 and ordered pairs: (25, F1), (−30, F2)

In order to convert the given values to degrees Fahrenheit, substitute the given values into the equation to get F1 and F2 respectively:

For (25, F1):

F = 9/5 C+32      (put C= 25 in the equation)

F1 = 9/5(25) + 32

F1 = 45 +22

F1 = 77 degrees Fahrenheit

For (−30, F2):

F = 9/5 C+32       (put C= -30 in the equation)

F2 = 9/5(-30) + 32

F2 = -54 + 32

F2 = -22 degrees Fahrenheit

Therefore, the values of F1 and F2 are 77 degrees Fahrenheit and -22 degrees Fahrenheit respectively

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The resulting equation is the x-coordinate of the point of local extrema of a quadratic function.

To find the local extrema of a polynomial function f(x), we apply the equation f'(x) = 0. We are given the basic form of a quadratic equation. The equation is given below.

f(x) = ax² + bx + c

We need to find the differentiation of the above equation.

f'(x) = 2ax + b

Now, we will equate the obtained equation to zero and solve for the value of the variable "x".

f'(x) = 0

2ax + b = 0

2ax = -b

x = -b/2a

This resulting equation is familiar from algebra/pre-calculus. This is the x-coordinate of the point of local extrema of a quadratic function.

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The cost of the 80 square inches metal sheet is $160

The length of the rectangular sheet = 8 inches

The width of the rectangular sheet = 10 inches

The area of  the rectangular sheet = The length of the rectangular sheet  × The width of the rectangular sheet

Substitute the values in the equation

= 8 × 10

= 80 square inches

The cost of metal per square inch = $2

Then the cost of metal sheet = The area of  the rectangular sheet × The cost of metal per square inch

Substitute the values in the equation

= 80 × 2

= $160

Hence, the cost of the 80 square inches metal sheet is $160

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Answer:

Step-by-step explanation: 32=v + 27

V= 5

how:

32-27=5

v=5

I don"t know what you meant for at the bottom, if that is included please explain what you mean.

Answer:

5

Step-by-step explanation:

32 = v + 27

Subtract 27 from both sides and make v the subject.

32 - 27 = v

5 = v

Answer:

120 inches by 80 inches

3 inches by 2 inches

Step-by-step explanation:

So the ratio of 12/8 is 1.5 so we divide all of the numbers to find ratios that equal that

120/80=1.5

3/2=1.5

but we dont put say 6 by 2 because 6/2 is 3

The third term from the sequence should be 48 so that (a) the next term is 147 (b) the rule to find the nth term is 3×n² and (c) the 20th term is 1200.

From the question, 3, 12, 27, 64, 75, 108 the third term 64 is supposed to be 48. hence the terms are derived as follows:

1st term; 3 × 1² = 3

2nd term;3 × 2² = 12

3rd term; 3 × 3² = 27

4th term; 3 × 4² = 48

5th term; 3 × 5² = 75

6th term; 3 × 6² = 108

Therefore, the next term which the 7th term is 3 × 7² = 147, the rule for the sequence is 3×n² and the 20th term is 3 × 20² = 1200.

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The slope is -3/(2+3a)² and the equation of the line is y - 1/8 = [-3/(2+3a)²](x - 2).

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

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

​f(x) = 1 / 2+3x and point P = (2 , 1/8)

f(x+h) = 1/(2 + 3a + 3h)

f(a) = 1/(2 + 3a)

m(tan) = [1/(2 + 3a + 3h) - 1/(2 + 3a)]/h

After simplifying:

m(tan) = -3/[(2+3a)(2+3a+3h)]

h = 0

m(tan) = -3/(2+3a)²

The equation of the line:

y - 1/8 = [-3/(2+3a)²](x - 2)

Thus, the slope is -3/(2+3a)² and the equation of the line is y - 1/8 = [-3/(2+3a)²](x - 2).

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Answer:

-1.25

Step-by-step explanation:

m=

[tex]y2 - y1 \div x2 - x1[/tex]

3- (-2)÷ -1-3

5/-4

m=-1.25

Answer:

Two correct answers: subtracting exponents and

3^ 2/9

Step-by-step explanation:

To divide, subtract exponents. To subtract fractions keep the bottom number the same and subtract the top number. See image.

Answer:

I hope this will be helpful for you.

0 (zero)

Step-by-step explanation:

[tex]{ \blue{ \sf{ {5}^{ - 3} \times {2}^{4} \times {5}^{3} \times {2}^{ - 5}}}} [/tex]

Arrange the like terms in order

[tex]{ \blue{ \sf{ {5}^{ - 3} \times {5}^{3} \times {2}^{4} \times {2}^{ - 5}}}} [/tex]

This is in the form of [tex]{ \red{ \sf{ {a}^{m} \times {a}^{n} = {a}^{m + n}}}} [/tex]

[tex]{ \blue{ \sf{ {5}^{ - 3 + 3} \times {2}^{4 + ( - 5)}}}} [/tex]

[tex]{ \blue{ \sf{ {5}^{0} \times {2}^{4 - 5}}}} [/tex]

[tex] { \blue{ \sf{0 \times {2}^{ - 1}}}} [/tex]

[tex]{ = \blue{ \sf{0}}}[/tex]

Any number which is multiplied with zero is zero itself.

Answer:

1. 5√7

2. (√15)/6

Step-by-step explanation:

1. √175

= √(25) * √(7)

= 5√7

2. √(5/12)

= (√5)/(√12)

= (√5)/(√(4) * √(3))

= (√5)/(2√3)

Radicals can't be in the denominator:

(√5)/(2√3)

(√5 * 2√3)/(2√3 * 2√3)

(2√15)/12

(√15)/6

The auxiliary function used in the composition between the two functions f(x) and g(x) is equal to linear function g(x) = x + 3.

Herein we find the result of a composition between two functions, where the variable x of the parent function f(x) is substituted by the auxiliary function g(x). The composition between the two functions is defined below:

h(x) = f ° g (x) = f [g (x)]

If we know that the parent function f(x) = x⁵, then the auxiliary function within the composition is the linear function g(x) = x + 3.

The statement is poorly formatted, the correct form is shown below: The function h(x) = (x + 3)⁵ can be expressed in the form f[g(x)] where f(x) = x⁵ and g(x) is defined below.

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Answer:

66 inches

Step-by-step explanation:

60 + .1(60)

60 + 6

66

Answer:

66 inches

Step-by-step explanation:

60 x .10 = 6

60 +6 =66

Answer: 14 is the ancer

Step-by-step explanation:

Answer:

its 14

Step-by-step explanation:

Answer:

  30 m

Step-by-step explanation:

Given a trapezium with area 756 m², height 28 m, and one base of length 24 m, you want to know the length of the other base.

The formula for the area of a trapezium is ...

  A = 1/2(b1 +b2)h

Using the given values, we have ...

  756 = 1/2(24 +b2)(28)

  54 = 24 +b2 . . . . . . . . . . . divide by 14

  30 = b2 . . . . . . . . . . . . subtract 24

The measure of the other parallel side is 30 m.

A linear function will have the form f(x) = mx + b.

f(0)=7 means the point (0,7) is on this line.

f(3)=1 means the point (3,1) is on the line as well.

Using those two points, we can find the slope:

   m = (1-7)/(3-0) = -6/3 = -2

Since we were given f(0)=7, we also know the y-intercept.

So our function is f(x) = -2x+7

Using the two points on the graph, Sandra can prove that her speed was constant using: Slope (m) = (18 - 6) / (3 - 1).

The slope of a graph is the ratio of the vertical distance along the line to the horizontal distance across the line, which is given as, m = change in y / change in x.

For a proportional graph, the slope is always constant, for whichever points we choose to use on the line.

Given two points on the line that Sandra chose, (3, 18) and (1, 6), therefore:

Slope (m) = change in y / change in x

Slope (m) = (18 - 6) / (3 - 1)

Slope (m) = 12/2

Slope (m) = 6

Therefore, to show her speed, which is the slope of the line, is constant, she can use the equation below:

Slope (m) = (18 - 6) / (3 - 1).

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Answer:

A ≈ 2.88×108km²

Step-by-step explanation:

A= 4 π r² = 4·π· 47902 ≈ 2.88324×10 power 8 km²

According to the information, it can be inferred that the turtle travels 0.5 miles in a quarter of an hour (1/4 hours).

To find the distance that the tortoise travels in a quarter of an hour we must perform the following operations and take into account the information provided.

To verify that the turtle takes 1/4 of an hour to travel 0.5 miles, we must divide the total number of miles it travels in one hour by 4, because we want to identify how much distance it travels in a quarter of the time, as shown below:

2 miles / 4 = 0.5 miles

Based on the above, we can infer that the tortoise travels 0.5 miles in 1/4 hour.

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Answer:

Step-by-step explanation:

You want the first 9 terms of the recursive sequence defined by ...

  [tex]\begin{cases}s_1=2\\s_2=3\\s_n=s_{n-2}(s_{n-1}-1)\end{cases}[/tex]

The attached spreadsheet uses the given formula for the next term. It shows the terms to be ...

  2, 3, 4, 9, 32, 279, 8896, 2481705, 22077238784

The attached spreadsheet sum function has been used to find the sum of the first 8 terms. That sum is ...

  2490930

The sequence of problem 1 has neither a common difference nor a common ratio between successive terms. It is neither arithmetic nor geometric.

The sum of the first n terms of an arithmetic sequence with first term a1 and common difference d is given by ...

  Sn = (2a1 +d(n -1))(n/2)

You have a sequence with a1 = 12 and d = (18-12) = 6. You want the sum of the first 200 terms.

  S200 = (2·12 +6(200 -1))(200/2) = 121,800

The sum is 121,800.

__

Additional comment

A spreadsheet is a nice tool for finding terms of a recursively-defined sequence. The formula can include as many terms as necessary, and it can be replicated thousands of times, if necessary. The limitation is that arithmetic is generally limited to 16 significant figures, or so.

Answer:

1.  

s₁ = 2

s₂ = 3

s₃ = 4

s₄ = 9

s₅ = 32

s₆ = 279

s₇ = 8,896

s₈ = 2,481,705

s₉ = 22,077,238,784

2. 2,490,930

3.  Neither.

4.  121,800

Step-by-step explanation:

A recursive rule for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Given recursive rule:

[tex]\begin{cases}s_n=s_{n-2} \cdot (s_{n-1}-1)\\s_1=2\\s_2=3\end{cases}[/tex]

Therefore, the first 9 terms of the sequence are:

[tex]s_1=2[/tex]

[tex]s_2=3[/tex]

[tex]\begin{aligned}s_3&=s_{3-2} \cdot (s_{3-1}-1)\\&=s_{1} \cdot (s_{2}-1)\\&=2 \cdot (3-1)\\&=2 \cdot 2\\&=4 \end{aligned}[/tex]

[tex]\begin{aligned}s_4&=s_{4-2} \cdot (s_{4-1}-1)\\&=s_{2} \cdot (s_{3}-1)\\&=3 \cdot (4-1)\\&=3 \cdot 3\\&=9 \end{aligned}[/tex]

[tex]\begin{aligned}s_5&=s_{5-2} \cdot (s_{5-1}-1)\\&=s_{3} \cdot (s_{4}-1)\\&=4 \cdot (9-1)\\&=4 \cdot 8\\&=32 \end{aligned}[/tex]

[tex]\begin{aligned}s_6&=s_{6-2} \cdot (s_{6-1}-1)\\&=s_{4} \cdot (s_{5}-1)\\&=9 \cdot (32-1)\\&=9 \cdot 31\\&=279\end{aligned}[/tex]

[tex]\begin{aligned}s_7&=s_{7-2} \cdot (s_{7-1}-1)\\&=s_{5} \cdot (s_{6}-1)\\&=32 \cdot (279-1)\\&=32 \cdot 278\\&=8896\end{aligned}[/tex]

[tex]\begin{aligned}s_8&=s_{8-2} \cdot (s_{8-1}-1)\\&=s_{6} \cdot (s_{5}-1)\\&=279 \cdot (8896-1)\\&=279\cdot 8895\\&=2481705\end{aligned}[/tex]

[tex]\begin{aligned}s_9&=s_{9-2} \cdot (s_{9-1}-1)\\&=s_{7} \cdot (s_{8}-1)\\&=8896\cdot (2481705-1)\\&=8896\cdot 2481704\\&=22077238784\end{aligned}[/tex]

Given series:

[tex]\displaystyle \sum^8_{k=1} s_k[/tex]

The sum notation asks to find the sum of the first 8 terms of the sequence from question 1.

Therefore:

[tex]\begin{aligned}\displaystyle \sum^8_{k=1} s_k&=s_1+s_2+s_3+s_4+s_5+s_6+s_7+s_8\\&=2+3+4+9+32+279+8896+2481705\\&=2490930\end{aligned}[/tex]

If a sequence is arithmetic, the difference between consecutive terms is the same (this is called the common difference).

If a sequence is geometric, the ratio between consecutive terms is the same (this is called the common ratio).

As the difference between consecutive terms it not the same, the sequence is not arithmetic.

As the ratio between consecutive terms is not the same, the sequence is not geometric.

Therefore, the sequence is neither arithmetic nor geometric.

[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of an arithmetic series}\\\\$S_n=\dfrac{1}{2}n[2a+(n-1)d]$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $d$ is the common difference.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given arithmetic sequence:

Therefore:

To find the sum of the first 200 terms, substitute the found values of a and d into the formula along with n = 200:

[tex]\begin{aligned}S_{200}&=\dfrac{1}{2}(200)[2(12)+(200-1)(6)]\\&=100[24+(199)(6)]\\&=100[24+1194]\\&=100[1218]\\&=121800\end{aligned}[/tex]

Answer:

acute angle - UVW

obtuse angle - VWX

right angle - UYX

There are 9216 more white squares than black squares in the 47th term with it's an odd number of 97

The terms are generated with the square of odd numbers with the black squares being the addition of the odd number and one subtracted from it.

For the 47th term:

Given that the first term is consists of 25 squares with 9 black and 16 white squares

first term has odd number 5

the square of 5; 5² = 25

black squares = 5 + (5-1) = 9

white squares = 25 - 9 = 16

Therefore, for the 47th term:

47th term has odd number 97

the square of 97; 97² = 9409

black squares = 97 + (97-1) = 193

white squares = 9409 - 193 = 9216

Hence, the 47th term has an odd number 97 with 9216 more black squares than the white squares,

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The difference of the given algebraic expressions is -2x²+12x+13.

The given algebraic expressions are -2x²+5x+10 and -7x-3.

To subtract an algebraic expression from another, we should change the signs (from '+' to '-' or from '-' to '+') of all the terms of the expression which is to be subtracted and then the two expressions are added.

The subtraction of given expressions is as follows

-2x²+5x+10-(-7x-3)

= -2x²+5x+10+7x+3

= -2x²+12x+13

Hence, the difference of the given algebraic expressions is -2x²+12x+13.

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Answer:

Step-by-step explanation: