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a) (i) Average velocity = -96.8 ft/s

   (ii) Average velocity = -95.2 ft/s

   (iii) Average velocity = -92.8 ft/s

b) The instantaneous velocity of the pebble after 3 seconds is approximately -96 ft/s.

What is average velocity?

The change in position or displacement (x) divided by the time intervals (t) in which the displacement happens yields the average velocity. Depending on the sign of the displacement, the average velocity can be positive or negative.

(a)

(i) The time period from t = 3 to t = 3.1 is 0.1 seconds. The average velocity of the pebble during this time period can be found by taking the difference in position (y) divided by the difference in time (t).

Average velocity = (y(3.1) - y(3)) / (3.1 - 3)

Substituting \(y = 225 - 16t^2\), we get:

\(Average\ velocity = [(225 - 16(3.1)^2) - (225 - 16(3)^2)] / (0.1)\)

Average velocity = -96.8 ft/s (rounded to one decimal place)

(ii) The time period from t = 3 to t = 3.05 is 0.05 seconds. The average velocity of the pebble during this time period can be found using the same formula as above:

\(Average\ velocity = (y(3.05) - y(3)) / (3.05 - 3)\)

Substituting \(y = 225 - 16t^2\), we get:

\(Average\ velocity = [(225 - 16(3.05)^2) - (225 - 16(3)^2)] / (0.05)\)

Average velocity = -95.2 ft/s (rounded to one decimal place)

(iii) The time period from t = 3 to t = 3.01 is 0.01 seconds. The average velocity of the pebble during this time period can be found using the same formula as above:

Average velocity = (y(3.01) - y(3)) / (3.01 - 3)

Substituting \(y = 225 - 16t^2\), we get:

\(Average \ velocity = [(225 - 16(3.01)^2) - (225 - 16(3)^2)] / (0.01)\)

Average velocity = -92.8 ft/s (rounded to one decimal place)

(b)

The instantaneous velocity of the pebble at any given time t can be found by taking the derivative of the position function y(t) with respect to time:

\(\frac{dy}{dt} =-32t\\\)

Substituting t = 3, we get:

\(\frac{dy}{dt} =-96ft/s\\\)

Therefore, the instantaneous velocity of the pebble after 3 seconds is approximately -96 ft/s.

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Answer:

1

Step-by-step explanation:

b= y intercept

Then 1 - 3/5 = 2/5. This means that 2/5 of the frogs in that particular part of the rainforest are not poisonous. This fraction can also be expressed as 40/100 or 0.4 in decimal form.

In the given scenario, we know that 3/5 of the frogs in a particular part of the rainforest are poisonous. This means that the remaining fraction of the frogs are not poisonous.

To find this fraction, we need to subtract 3/5 from 1 (since the sum of the fractions of poisonous and non-poisonous frogs is equal to 1).

It's important to note that just because a frog is not poisonous, it doesn't mean it's safe to touch or handle them. It's always best to admire these beautiful creatures from a safe distance and leave them alone in their natural habitat.

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The impact on Carns Company's income from eliminating the Small Tools Division would be a decrease of $1,209,000.

To determine the impact on Carns Company's income from eliminating the Small Tools Division, we need to consider the avoidable costs associated with the division.

The avoidable costs include all of the variable costs of the division and a portion of the fixed costs that are specifically related to the Small Tools Division. In this case, the variable costs of the division are $1,295,000, and $119,000 of the fixed costs are avoidable.

To calculate the impact on income, we subtract the avoidable costs from the loss reported by the division:

Impact on income = Loss - Avoidable costs

Impact on income = $205,000 - ($1,295,000 + $119,000)

Impact on income = $205,000 - $1,414,000

Impact on income = -$1,209,000

The negative sign indicates a decrease in income.

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The computation of the increased in salary is $500.

Here, we have,

Explanation:

Data provided in the question

Salary for the first year = $50,000

CPI increase during the year = 4%

Overstated inflation = 1% i.e. 5%

The computation of the increased in salary is shown below:

= Salary of the first year × inflation rate - salary of the first year × CPI increase during the year

= $50,000 × 5% - $50,000 × 4%

= $2,500 - $2,000

= $500

Hence, The computation of the increased in salary is $500.

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complete question:

Mark's wage contract specifies a $50,000 salary for the first year, and specifies a salary increase equal to the percentage increase in the CPI during the second year. The increase in the CPI during the year was 4.0%. If the CPI overstates inflation by 1.0% (that is, the actual price increase was 3% and not 4%), at the end of the first year Mark's salary increased by $________ more than it would have without the upward bias.

Answer:

-22=22

Step-by-step explanation:

3,5-5,7=

-22/22

The equation of the line passing through R and PQ is 4(y - 6) = -x - 1/2.

To find the equation of the line passing through the midpoint R and the points P and Q, we first need to find the coordinates of the midpoint R. The midpoint coordinates can be found by taking the average of the x-coordinates and the average of the y-coordinates of P and Q.

The x-coordinate of the midpoint R is (3 + (-5)) / 2 = -1/2.

The y-coordinate of the midpoint R is (5 + 7) / 2 = 6.

So, the coordinates of the midpoint R are (-1/2, 6).

Next, we can use the two-point form of the equation of a line, which states that the equation of the line passing through points (x₁, y₁) and (x₂, y₂) is given by:

(y - y₁) = (y₂ - y₁) / (x₂ - x₁) \(\times\) (x - x₁)

Substituting the coordinates of R (-1/2, 6) and P (3, 5) into the equation, we have:

(y - 6) = (7 - 5) / (-5 - 3) \(\times\)(x - (-1/2))

Simplifying the equation:

(y - 6) = (2 / -8) \(\times\)(x + 1/2)

(y - 6) = -1/4 \(\times\)(x + 1/2)

4(y - 6) = -x - 1/2

Therefore, the equation of the line passing through R and PQ is 4(y - 6) = -x - 1/2.

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Answer: 46

Step-by-step explanation:

Yes !!

Answer: cos4

Step-by-step explanation: 50 meters

B. 50-√3 meters

C. 100 meters

D. 100-√3 meters

40. What expression is used to answer: "A 4-m tall man stands on horizontal ground 43 m from a tree.

The angle of depression from the top of the tree to his eyes is 22°. Estimate the height of the tree."

A. sin 22

B. cos 22

C. tan 22

D. cot 22

THE ANSWER IS ###3 My gUy ! 1 ! ! !1 !1!

Answer:

The answers is 3

Step-by-step explanation:

Answer:

Step-by-step explanation:

Mustafa's soccer team is planning a school dance as a fundraiser. The DJ charges $200 and decorations cost $100.

The team decides to charge each student $5.00 to attend the dance. If n represents the number of students attending

the dance, which equation can be used to find the number of students needed to make $1,500 in profit?

Answer:

Step-by-step explanation:

This article is about the basic inequality {\displaystyle z\leq x+y}{\displaystyle z\leq x+y}. For other inequalities associated with triangles, see List of triangle inequalities.

Three examples of the triangle inequality for triangles with sides of lengths x, y, z. The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y.

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.[1][2] This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality.[3] If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that

{\displaystyle z\leq x+y,}z\leq x+y,

with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):

{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}\|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,

Answer:

x=−2−√5 or x=−2+√5

Step-by-step explanation:

-5x2 - 20x + 35 = 30

Answer = x=−2−√5 or x=−2+√5

The coordinates of Kyle's house would be (x - 4, y + 2) relative to Luke's house, where x and y represent the Coordinates of Luke's house on the coordinate grid.

The grid representing Kyle's house. However, based on the given information, we can determine the coordinates of Kyle's house relative to Luke's house.

If Luke's house is located at the point (x, y), then Kyle's house is located 4 units to the left of Luke's house, which means the x-coordinate of Kyle's house is (x - 4). Similarly, Kyle's house is located 2 units up from Luke's house, which means the y-coordinate of Kyle's house is (y + 2).

Therefore, the coordinates of Kyle's house would be (x - 4, y + 2) relative to Luke's house, where x and y represent the coordinates of Luke's house on the coordinate grid.

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A. The following are sets A and B:

A = {2, 3, 5, 7, 11}

B = {1, 2, 3, 4, 6, 12}

C. Elements not in A or A' = ∅

B. The Venn diagram is attached

A universal set is a set which consists of all elements related to the given sets. It is denoted by U.

A. Set A:

A = {2, 3, 5, 7, 11}

Set B:

B = {1, 2, 3, 4, 6, 12}

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A' = {1, 4, 6, 8, 9, 10, 12}

C. Elements not in A or A' = ∅

Complement of set A is refers to a set that contains the elements present in the universal set but not in set A.

Hence, the Venn diagram of sets A, B and U has been attached.

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Note that  the new triangle will have a base of 10 inches and the same side lengths as the given triangle (13 inches).

The perimeter of the new triangle is the sum of all three sides is 36 inches.

To construct another isosceles triangle with the same area as the given triangle, we can use the formula for the area of a triangle  -  

A =  (base x height)/2

In the   given triangle,the base is 10 inches.

Let's assume the height of the given triangle is h.

The area of the given triangle is

A = (1/2) * 10 * h

= 5h in²

To construct another isosceles triangle with the same area, we need to find a different base, let's call it x, such that

(1/2) * x * h = 5h.

Simplifying the equation, we get

x = 10.

Therefore, the new triangle will have a base of 10 inches and the same side lengths as the given triangle (13 inches).

The perimeter of the new triangle is the sum of all three sides: 10 + 13 + 13 = 36 inches.

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Answer:

x<-12 or x>16

Step-by-step explanation:

Answer:

|x - 2| + 3 > 17

|x - 2| > 14

x - 2 < -14 or x - 2 > 14

x < -12 or x > 16

Answer:

first piece is 2 inches second piece is 4 inches third piece is 6 inches

Step-by-step explanation:

x=2

1(2) + 2(2) + 2(3)

Answer:

Step-by-step explanation:

Let h(x) = y

The exponentail function is of the form :

\(y = ab^x\)

We have :

\(y_{_1} = ab^{x_{_1}}\\y_{_2} = ab^{x_{_2}}\\\\\implies \frac{y_{_1}}{y_{_2}} = \frac{ab^{x_{1}}}{ab^{x_{2}}} \\\\\implies \frac{y_{_1}}{y_{_2}} = \frac{b^{x_{1}}}{b^{x_{2}}} \\\\\implies \frac{y_{_1}}{y_{_2}} = b^{(x_1-x_2)}\)

Given points : (4, 9) and (5, 34.2)

We have:

\(\frac{34.2}{9} = b^{(5-4)}\\\\\implies 3.8 = b\)

Writing the equation with x, y and b:

\(y = ab^x\\\\\implies 9 = a(3.8^4)\\\\a = \frac{9}{3.8^4} \\\\a = 0.043\)

a = 0.043

b = 3.8

When x = 6, y will be:

\(y = (0.043)(3.8^6)\\\\y = 128.47\)

This is not the y value in the question y = 59.4

Therefore (6, 59.4) does not lie on the graph h(x)

The equation x² - 6x = 72 has a solution between 4 and 5

Use a trial and improvement method to find this solution.

There are a number of "improvement" techniques that can be used to improve an estimate of a solution to a polynomial. Some of them use the polynomial itself to form the iterator function.

Iterator

There are a number of ways we can solve the given equation for x. Often, at least one of these will converge when used for improving the estimated value of x.

 x₁ = (\(x^{2}\) -72)/6 . . . . . . . add 6x-72, divide by 6

 x₂ = 72/(x^2 -6) . . . . . . . factor out x and divide by its coefficient

 x₃ = ∛(72 +6x) . . . . . . . add 6x and take the cube root

The first two of these iterators tend to diverge. For x = 4.5, they give ...

 x₁ = (4.5³ -72)/6 = 3.1875

 x₂ = 72/(4.5² -6) = 5.0526

Both of these stray out of the given interval of [4, 5].

However, the third one tends to converge, so can give us a useful approximation in one iteration:

 x₃ = ∛(72 +6×4.5) ≈ ∛99 ≈ 4.6261

Second iteration

Applying this again, we get ...

 x ≈ ∛(72 +6·4.6261) ≈ 4.6378

The approximate root to 1 decimal place is 4.6.

Additional comment

One of the quickest converging iterators is that provided by Newton's method. For finding the solution to f(x) = 0, it uses ...

 x₁ = x -f(x)/f'(x) . . . . . . . where f'(x) is the first derivative

For this problem, we can define ...

 f(x) = \(x^{2}\) -6x -72   ⇒   f'(x) = 3x^2 -6

 x₁ = x -(\(x^{2}\) -6x -72)/(3x^2 -6) = (3\(x^{2}\) -6x -\(x^{2}\) +6x +72)/(3x^2 -6)

 x₁ = (2\(x^{2}\) +72)/(3x^2 -6)

Then for x = 4.5, the "improved" solution is ...

 x₁ = (2(4.5³) +72)/(3(4.5²) -6)) = 254.25/54.75 = 339/73 ≈ 4.6438

A second iteration gives ...

 x ≈ 4.639026 . . . . accurate to 5 decimal places

Hence with this iterator, the number of accurate decimal places tends to double with each iteration.

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Answer:

8/3

Step-by-step explanation:

Answer:

3 1/3

Step-by-step explanation:

sorry

Splitting the rectangle in half creates two right triangles

The value of tan(x) is 4/3

To calculate tan(x), we make use of the following tangent ratio

tan(x) = Opposite/Adjacent

From the figure, we have:

Opposite = AD = 4

Adjacent = CD = 3

So, we have:

tan(x) = 4/3

Hence, the value of tan(x) is 4/3

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Answer:

4/3

Step-by-step explanation:

Answer:

x=121

Step-by-step explanation:

1. Move the constant to the right-hand side and change its sign

square rootx = 6+5

2. Add the numbers

square rootx= 11

3. Square both sides of the equation

x=121

Check if the given value is the solution of the equation

square root 121 -5 =6

Simplify the expression

6=6

The equality is true, therefore x=121 is a solution of the equation

Answer:

A. 76

B. x^2 +21x + 96

C. x^2 - 9x + 6

Step-by-step explanation:

For this problem, just plug in the number in f(x) for x in the function.

A. f( 5 ) -> Plug in 5 for every x and then simplify

(5)^2 + 9(5) + 6

= 25 + 45 + 6

= 76

B. f( x+6 ) -> Plug in (x+6) for every x and then simplify

(x+6)^2 + 9(x+6) + 6

x^2 + 12x +36 + 9x+ 54 + 6

x^2 +21x + 96

C. f( -x ) -> Plug in -x for every x and then simplify

(-x)^2 + 9(-x) + 6

x^2 - 9x + 6

Answer:

Step-by-step explanation:

Given a plane passing through the point(1, 5,-1) and perpendicular to the vector (1, 5, 1), the equation of the plane can be expressed generally as;

a(x-x₀)+b(y-y₀)+c(z-z₀) = 0 where (x₀, y₀, z₀) is the point on the plane and (a, b,c) is the normal vector perpendicular to the plane i.e (1,5,1)

Given the point P (1, 5, -1) and the normal vector n(1, 5, 1)

x₀ = 1, y₀ =5, z₀ = -1, a = 1, b = 5 and c = 1

Substituting this point in the formula we will have;

a(x-x₀)+b(y-y₀)+c(z-z₀) = 0

1(x-1)+5(y-5)+1(z-(-1)) = 0

(x-1)+5(y-5)+(z+1) = 0

x-1+5y-25+z+1 = 0

x+5y+z-1-25+1 = 0

x+5y+z-25 = 0

x+5y+z = 25

The final expression gives the equation of the plane.

Answer:

Like Terms: Terms that have identical variable parts (same variable(s) and same exponent(s)). When simplifying using addition and subtraction, you combine “like terms” by keeping the "like term" and adding or subtracting the numerical coefficients.

Step-by-step explanation:

Answer:

events at A and B are independent because P(A)(B)≠P(A)

Answer:

14 √2

Step-by-step explanation:

 Given expression:

             √392

Simplified radical = ?

Solution:

      392  = 8 x 49

    =  √8 x 49

    =   7√8

    =   7√4 x 2

    = 7 x 2√2

    = 14 √2

A. x² - 200 gives the amount of money the interior designer will make decorating the house as a function of hours he works.

B. He will earn 1,875 if he works for 50 hours.

C. The difference quotient is equal to 2x.

The solution has been obtained by using functions.

What is a function?

Function is a kind of relation, or rule, that associates a single input with a single, predetermined output.

We are given two functions

f(x) = 49x² – 200

which represents the amount of money he earns per room decorated

g(x) = (1/7)x

which represents the number of rooms the interior designer decorates

A. The amount of money the interior designer will make decorating the house as a function of hours he works will be given by evaluating f(x) in g(x).

So,

f(g(x)) = 49((1/7)x)² – 200

f(g(x)) = x² - 200

B. Since, the house requires 50 hours of work, this means that x = 50.

Substituting this in f(g(x)) = x² - 200, we get

f(g(50)) = 50² - 200

f(g(50)) = 1,875

C. The difference quotient for the function is as follows

\(\lim_{h \to \ 0}\frac{f(x+h) -f(x)}{h}\)

\(\lim_{h \to \ 0}\frac{(x+h)^{2}-200-x^{2}+200}{h}\)

\(\lim_{h \to \ 0}\frac{x^{2} + 2xh + h^{2}-x^{2}}{h}\)

= 2x

Hence, x² - 200 gives the amount of money and he earns 1,875 if he works for 50 hours. The difference quotient is equal to 2x.

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The annual interest rate is 7.5%.

What is Simple interest?

Simple interest is a type of interest that is calculated on the principal amount (or the initial amount) of a loan or investment, without taking into account any accumulated interest from previous periods.

The formula for simple interest is:

I = P × r × t

where I is the interest earned, P is the principal (initial amount), r is the annual interest rate as a decimal, and t is the time in years.

In this case, we know I = $562.50, P = $1500, and t = 5 years.

we ca substitute values into the formula,

562.50 = 1500× r ×5

Simplifying, we get:

r = 562.50 / (1500 ×5)

r = 0.075

r = 7.5%

Therefore, the annual interest rate is 7.5%.

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