a. Find the derivative function f' for the function f.
b. Determine an equation of the line tangent to the graph of f at​ (a,f(a)) for the given value of a.

The proportionality constant is the ratio that connects two given values in a proportional relationship. The constant of proportionality is also known as the constant ratio, constant rate, unit rate, constant of variation, or even the rate of change.

When two or more parameters are directly or indirectly proportional to one another, their relationship is expressed as a = kb or a = k/b, where k indicates the nature of the relationship between the two variables. This k is referred to as the proportionality constant.

For example, the number of apples in a crop is proportional to the number of trees in an orchard, with the ratio of proportionality being the average number of apples per tree.

This can be illustrated as:

Let's say y = 4

Let's say x = 2

y = kx

4 = 2k

Divide

k = 4/2

k = 2

The constant is 2.

Note that an overview was given as the information is incomplete.

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Answer:

feeling good

Step-by-step explanation:

I am just happy

Answer: x = -2, y = 3

Step-by-step explanation:

Given that,

-9x+2y = 24..................................(i)

x = 10-4y.......................................(ii)

From equation (i)

-9(10-4y) + 2y = 24  [as x = 10-4y]

=> -90 +36y + 2y = 24

=> 38y = 90 + 24

=> 38y = 114

=> y = 114/38

Therefore, y = 3

Again,

x = 10 - 4(3)  [as y = 3]

=> x = 10-12

Therefore, x = -2

Here, x = -2 and y = 3

Answer:

The three numbers are -4, 0, and 1.

The sum of three numbers is -3: -4 + 0 + 1 = -3

If the second number is subtracted from the sum of the first and third numbers, the result is -3: (-4 + 1) - 0 = -3

If the third number is subtracted from the sum of the first and second numbers, the result is -5: (-4 + 0) - 1 = -5

Step-by-step explanation:

x --> first number

y --> second number

z --> third number

x + y + z = -3

(x + z) - y = -3

(x + y) - z = -5

x = -3 - y - z

(-3 - y - z + z) - y = -3

-3 - y - y = -3

-2y = 0

y = 0

(-3 - y - z + y) - z = -5

-3 - z - z = -5

-3 - 2z = -5

-2z = -2

z = 1

x = -3 - 0 - 1

x = -4

No,  the slope of a horizontal line y=6 is,  zero not negative.

What is a Slope?

In Mathematics, a slope of a line is the change in y coordinate with respect to the change in x coordinate.

The net change in y-coordinate is represented by Δy and the net change in x-coordinate is represented by Δx.

Hence, the change in y-coordinate with respect to the change in x-coordinate is given by,

m = change in y/change in x = Δy/Δx

Where “m” is the slope of a line.

Generally, the slope of a line gives the measure of its steepness and direction. The slope of a straight line between two points says (x1,y1) and (x2,y2) can be easily determined by finding the difference between the coordinates of the points. The slope is usually represented by the letter ‘m’.

Slope Formula

If P(x1,y1) and Q(x2,y2) are the two points on a straight line, then the slope formula is given by:

Slope, m = Change in y-coordinates/Change in x-coordinates = Δy/Δx

m = (y2 – y1)/(x2 – x1)

In the given equation of line y=6 .

The line crosses the y-axis at 6, as the line is always at that value. As it is a horizontal line, the slope is zero.

The equation y = 6 means for each and every value of x, y will be equal to 6.

This is the definition of a horizontal line.

By definition, the slope of a horizontal line is m = 0

When x = 0 then y = 6

Therefore, the y-intercept is: (0, 6).

Hence , the the slope of a horizontal line , m = 0.

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Answer:

1st = 2 sin (x)

2nd = cos^2 (x)

3rd = sin^2 (x)

Answer:

Step-by-step explanation:

Take the square root of both sides

[tex]\sqrt{x^2}[/tex]=[tex]\sqrt{25}[/tex]

x=±5

Using Laws of exponents, the solution to the expression √(3p⁹q⁷r⁶)/(16x⁶y¹²) is;  (p⁴q³r³/(x³y⁶))√(3pq/16)

The different laws of exponents are;

1) Product With the Same Bases; a^m × a^n = a^(m+n)

where;

m and n are real numbers.

2) Quotient with Same Bases;

a^m/a^n = a^(m-n)

where;

a is a non-zero term and m and n are integers.

3) Power Raised to a Power; (a^m)^n = a^(mn)

where;

a is a non-zero term and m and n are integers.

4) Product to a Power; a^n * b^n = (ab)^(n)

where;

a is a non-zero term and n is the integer.

5) Quotient to a Power; It states that the fraction of two different bases with the same power is represented as; a^n/b^n = (a/b)^n

where;

a and b are non-zero terms and n is an integer.

6) Zero Power; When the power of any integer is zero, then it means that its' value is equal to 1, such that;

a⁰ = 1

where;

a is any non-zero term.

7) Negative Exponent Rule; a^(-m) = 1/(a^m)

We are given the expression as;

√(3p⁹q⁷r⁶)/(16x⁶y¹²)

Applying the laws of exponents, we will have;

√(3p⁹q⁷r⁶)/(16x⁶y¹²) = (p⁴q³r³/(x³y⁶))√(3pq/16)

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Answer:

it would be $3.02

Step-by-step explanation:

you pretty much do $6.04/2 Ib

therefore, your answer would be $3.02

Answer:

Step-by-step explanation:

6.04 divded by 2= 3.02 per pound

Answer:

Step-by-step explanation:

x=7/11

Answer:

Step-by-step explanation:

Slope: 6/3 = -2

since the line is going from left to right, it is a negative slope

Answer:

-7/3 or 2.3333333333333

Step-by-step explanation:

X= -1 – -4 = 3

Y= -4 – 3 = -7

y = -2.3333333333333x – 6.3333333333333

or

y = -7/3 x - 19/3

So when y= -6.33 that will mean x= -2.71

The economy's nominal GDP in 2006 and real GDP in 2010 is option B)  Nominal GDP in 2006 $1,000, Real GDP in 2010 = $2,000.

What does GDP Price Index mean?

Nominal GDP in 2006=$1,000;

Real GDP in 2010=$2,000

Price index = (nominal GDP/real GDP) ×100

thus, nominal GDP in 2006=(price index × real GDP)/100

= (100×$1,000)/100

=$1,000

Real GDP in 2010= (nominal GDP/price index) ×100

= ($2,200/110) ×100

=$2,000.

Hence, the economy's nominal GDP in 2006 and real GDP in 2010 is option B)  Nominal GDP in 2006 $1,000, Real GDP in 2010 = $2,000.

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If Robert bought a used car for $10,332 with an added sales tax of 2.5%. Then the original price of the car would be around $10,080.

Option (B) is correct.

What is a percentage?

A percentage is a relative value that represents the hundredth part of any quantity. One percent (symbolized 1%) is the one-hundredth part; thus, 100 percent represents the entire amount and 200 percent specifies twice the given amount.

Robert bought a used car for $10,332 with an added sales tax of 2.5%.

so the additional amount would be = 10,332 * 2.5%

                                                           = 10332 * 0.025

                                                           = $258.3

So the original price of the car would be = 10332 - 258.3

                                                                    ≈ $10,080.

Hence, If Robert bought a used car for $10,332 with an added sales tax of 2.5%. Then the original price of the car would be around $10,080.

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The relation among unit rate, slope and constant rate of change of a proportional linear relationship is y = kx.

What is unit rate?

The rate of change is the ratio between the x and y (or input and output) values in a relationship. Another term for the rate of change for proportional relationships is the constant of proportionality. If the rate of change is yx, then so is the constant of proportionality.

A unit rate means a rate for one of something. We write this as a ratio with a denominator of one.

The slope is defined as y = mx +b

b =1, m=k

y =k x.

Here, k is constant rate of change.

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B.

The age of a person's systolic pressure is 130 mm Hg is 45.98 years.

P = 0.004[tex]y^{2}[/tex] - 0.01y + 122

substitute p = 130 mm

130 = 4/1000 [tex]y^{2}[/tex] - 1/100 y + 122

8 = 4[tex]y^{2}[/tex] - 10y / 1000

8000 = 4[tex]y^{2}[/tex] - 10 y

[tex]2y^{2} -5y -4000=0[/tex]

y = 45.98 years

Therefore the age of a person's systolic pressure is 130 mm Hg is 45.98 years.

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Answer: It could be (3, -3) (3,3) or (-3,-3)

Step-by-step explanation:

they are all 3 away.

The distance between point (12,6) and the line y = x - 14 is given as follows:

5.6 feet.

The standard equation of a line is given by the rule presented as follows:

ax + by + c = 0.

The coordinates of the point which we want to find the distance from the line are given as follows:

(x0,y0).

The equation that obtains the shortest distance of the point (x0, y0) to the line ax + by + c = 0 is presented as follows:

[tex]D = \frac{|a(x_0) + b(y_0) + c|}{\sqrt{a^2+b^2}}[/tex]

The line for this problem is given by the rule presented as follows:

y = x - 14.

In standard notation, it is defined by the rule presented as follows:

x - y - 14 = 0.

Hence the coefficients are presented as follows:

a = 1, b = -1, c = -14.

The point has the coordinates given as follows:

(12,6).

Hence:

x0 = 12, y0 = 6.

Thus the distance from the point to the line is obtained as follows:

[tex]D = \frac{|a(x_0) + b(y_0) + c|}{\sqrt{a^2+b^2}}[/tex]

D = |1(12) -1(6) - 14|/sqrt(1²+(-1)).

D = 8/sqrt(2)

D = 5.6 feet.

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The equation to represent this scenario is y = 43*(3)ˣ  , the correct option is (a) .

In the question ,

it is given that ,

Number of pennies that Kyle starts the fundraiser =  43 pennies

goal is to triple the number of pennies each day .

let  "x" to represent the number of days .

let the total number of pennies be "y" .

this situation is represented by an exponential increase ,

that can be represented as

y = 3ˣ * 43

y = 43*3ˣ

Therefore , The equation to represent this scenario is y = y = 43*3ˣ  .

The given question is incomplete , the complete question is

Kyle starts a fundraiser with 43 pennies. He recruits all of the players on his baseball team to participate. His goal is to triple the number of pennies each day. Assume Kyle meets his daily goal. Use x to represent the number of days and y to represent the total number of pennies .

Select the Equation that represents the this scenario ?

(a) y = 43(3)ˣ

(b) y = 3(43)ˣ

(c) y = [(43)(3)]ˣ

(d) y = 43(x)³

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The  amount of money that she yas to pay at the end of 3 yesrs to clear her debt  is:  $16,140.

First step is to find the interest

Interest = ( Principal × Interest rate × time)

Interest = $12,000 × 11.5% × 3

Interest = $4,140

Now let find the amount she use to clear the debt

Amount = Principal + Interest

Amount = $12,000 + $4,140

Amount = $16,140

Therefore she will use $16,140 to clear the debt.

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point a and point e are the two points representing extreme points on graph

Answer:

a) Consecutive interior angles.

b) Alternate exterior angles.

c) Corresponding angles.

d) Alternate interior angles.

e) Corresponding angles.

f) None.

g) Alternate exterior angles.

h) Consecutive interior angles.

i) None.

j) Corresponding angles.

k) Corresponding angles.

l) Alternate exterior angles.

m) Alternate interior angles.

n) Consecutive interior angles.

Step-by-step explanation:

Transversal

A line that crosses two other lines in the same plane at two distinct points.

Corresponding angles

A pair of angles that are in the same position relative to both lines crossed by the transversal.

Alternate interior angles

A pair of angles on the inner side of the two lines that the transversal crosses, but on opposite sides of the transversal.

Alternate exterior angles

A pair of  angles on the outer side of the two lines that the transversal crosses, but on opposite sides of the transversal.

Consecutive interior angles

A pair of angles on the inner side of the two lines that the transversal crosses, and are on the same side of the transversal.

a) Lines p and q are crossed by transversal line s.

Therefore, ∠4 and ∠7 are consecutive interior angles as they are on the inner side of lines p and q, and are on the same side of the transversal s.

b) Lines r and s are crossed by transversal line p.

Therefore, ∠2 and ∠11 are alternate exterior angles as they are on the outer side of lines r and s, but on opposite sides of the transversal p.

c) Lines p and q are crossed by transversal line s.

Therefore, ∠12 and ∠16 are corresponding angles as they are in the same position relative to lines p and q crossed by the transversal s.

d) Lines r and s are crossed by transversal line q.

Therefore, ∠8 and ∠13 are alternate interior angles as they are on the inner side of lines r and s, but on opposite sides of the transversal q.

e) Lines p and q are crossed by transversal line s.

Therefore, ∠11 and ∠15 are corresponding angles as they are in the same position relative to lines p and q crossed by the transversal s.

f) ∠7 and ∠10 have no relationship.

g) Lines p and q are crossed by transversal line r.

Therefore, ∠1 and ∠14 are alternate exterior angles as they are on the outer side of lines p and q, but on opposite sides of the transversal r.

h) Lines p and q are crossed by transversal line s.

Therefore, ∠12 and ∠15 are consecutive interior angles as they are on the inner side of lines p and q, and are on the same side of the transversal s.

i) ∠6 and ∠7 have no relationship

j) Lines r and s are crossed by transversal line p.

Therefore, ∠1 and ∠3 are corresponding angles as they are in the same position relative to lines r and s crossed by the transversal p.

k) Lines r and s are crossed by transversal line q.

Therefore, ∠14 and ∠16 are corresponding angles as they are in the same position relative to lines r and s crossed by the transversal q.

l) Lines r and s are crossed by transversal line q.

Therefore, ∠6 and ∠15 are alternate exterior angles as they are on the outer side of lines r and s, but on opposite sides of the transversal q.

m) Lines p and q are crossed by transversal line r

Therefore, ∠5 and ∠10 are alternate interior angles as they are on the inner side of lines p and q, but on opposite sides of the transversal r.

n) Lines r and s are crossed by transversal line q.

Therefore, ∠8 and ∠14 are consecutive interior angles as they are on the inner side of lines r and s, and are on the same side of the transversal q.

Answer:

She will pay the same amount at both

Step-by-step explanation:

For Life Studio,

the initial sign up fee + $30 each month

= $30 = 30 × 6 = $180

Hence, $60 + $180 = $240

For Yoga Mind,

$40 each month for 6 months

= $40 × 6

$40 × 6= $240

Therefore, she will pay the same amount at both

Answer:

36

48

60

Reason:

Since it is multiples of twelve, we can find out and multiply

12*3= 36

12*4= 48

12*5= 60

I hope this helps!

Answer: 36, 48, and 60.

Step-by-step explanation:

36, 48, and 60 are the only numbers that can be divided by 12 into a whole number.

36/12 = 3

48/12 = 4

60/12 = 5

Mazda was traveling faster than Ford Explorer by 0.57 miles per hour.

We know that the relation between distance, speed, and time is given by:

distance = speed * time

speed = distance / time

We are given;

A Mazda traveled 326 miles in 7 hours.

A Ford Explorer traveled 230 miles in 5 hours.

Let us find the rate or speed one by one:

For Mazda;

Distance = 326 miles

Time = 7 hours

Rate = distance / time = 326 / 7 mile per hour = 46.57 mile per hour

For Ford Explorer;

Distance = 230 miles

Time = 5 hours

Rate = distance / time = 230 / 5 mile per hour = 46 mile per hour

We can analyze that Mazda was traveling faster than Ford Explorer.

Thus, Mazda was traveling faster than Ford Explorer by 0.57 miles per hour.

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If a decimal number, x = 0.3 then 10x is 10 multiplying 0.3 the result is 3

A decimal is a number that is divided into two parts: a whole and a fraction.

Between integers and fractions, decimal numbers reflect the numerical value of whole

0.3 is x which is a decimal

10 multiplies 0.3 the result is 3

The process involve converting a fraction to a whole number as the result of multiplication changed the number system from decimal to a whole number

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Jon's semi-final jump compared with Anthony's was 3 inches lower. This was based on the heights given.

Given that Jon jumped 1 inches below the qualifying height, but his friend Anthony

made it to 2 inches above the qualifying height.

In this case, the difference in the jump will be gotten by subtracting the values. This will be illustrated as:

= Anthony's jump - Jon's jump

= 2 - (-1)

= 2 + 1

= 3

Therefore, Jon was 3 inches lower.

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The rate of the boat in still water is 45.5 km/hr and  the rate of the current is 100.5 km/hr.

A motorboat travels 74 km in 2 hours going upstream.

Rate = Distance/time

Rate = 74 /2

Rate = 36 km/hr

It travels 110 km going downstream

Rate = Distance/time = 110/2

Rate  = 55 km/hr

Equations:

Upstream: b-c = 36 km/hr

Downstrm: b+c = 55 km/hr

Add and solve for b

2b = 36 km/hr + 55 km/hr

2b = 91 km/hr

Divide both side by 2b

b = 91 km/hr /2

b = 45.5 km/hr (speed of the boat)

Solve for c

b+c = 55

45.5+ c = 55

c = 55 + 45.5

c = 100.5 km/hr (speed of the current)

Therefore  45.5 km/hr  and  100.5 km/hr are the speed.

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The solutions of the inequality are given by:

x > -3

Here we have the following inequality:

-(4x - 5) + 16x > -31

To solve this, we need to isolate the variable x in one of the sides of the inequality, so let's do that.

-(4x - 5) + 16x > -31

-4x + 5 + 16x > -31

(-4 + 16)*x + 5 > -31

12x + 5 > -31

12x > -31 - 5

12x> -36

x > -36/12

x > -3

So all the values of x that are larger than -3 are solutions of the inequality.

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The equation of the horizontal line passing through the point (2, -6) is :

y = -6

and the equation of the vertical line passing through the point (2, -6):

x = 2

We need to write equations for the horizontal and vertical lines passing through the point (2, -6)

For a vertical line the value of x will never change. This will be true for any value of y.

So, the equation for a vertical line passing through this point is:

x = 2

Similarly, for a horizontal line the value of y will never change. This is true for any value of x.

So, the equation for a horizontal line passing through this point is:

y = -6

Therefore, the equation of the horizontal line passing through the point (2, -6) is y = -6

and the equation of the vertical line passing through the point (2, -6) is x = 2

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