Point B is between A and C. AC = 42, BC = x + 12, and AB = x +6.
Find the value of x.

The explicit formulas for the compositions of functions are as follows: (a) f o g: (2(5x + 7)) + 3 = 10x + 17, (b) g o f: (5(2x + 3)) + 7 = 10x + 22, (c) f o h: 2(x^2 + 1) + 3 = 2x^2 + 5, and (d) h o f: (2x + 3)^2 + 1 = 4x^2 + 12x + 10.

To find the explicit formulas for the compositions of functions, we substitute the inner function into the outer function and simplify the expression.

(a) f o g:

Substituting g(x) = 5x + 7 into f(x) = 2x + 3, we get f(g(x)) = f(5x + 7) = 2(5x + 7) + 3 = 10x + 17.

(b) g o f:

Substituting f(x) = 2x + 3 into g(x) = 5x + 7, we get g(f(x)) = g(2x + 3) = 5(2x + 3) + 7 = 10x + 22.

(c) f o h:

Substituting h(x) = x^2 + 1 into f(x) = 2x + 3, we get f(h(x)) = f(x^2 + 1) = 2(x^2 + 1) + 3 = 2x^2 + 5.

(d) h o f:

Substituting f(x) = 2x + 3 into h(x) = x^2 + 1, we get h(f(x)) = h(2x + 3) = (2x + 3)^2 + 1 = 4x^2 + 12x + 10.

Therefore, the explicit formulas for the compositions of functions are: (a) f o g: 10x + 17, (b) g o f: 10x + 22, (c) f o h: 2x^2 + 5, and (d) h o f: 4x^2 + 12x + 10.

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The number of different routes he can take is 1287.

Given:

A jogger jogs every morning to his health club, which is eight blocks east and five blocks north of his home. He always takes a route that is as short as possible, but he likes to vary it .

Total blocks = 8+5 = 13

Number of different routes = \(13_C_5\)

= 13!/5!(13-5)!

= 13!/5!8!

= 13*12*11*10*9*8!/5!8!

= 13*12*11*10*9 / 5*4*3*2*1

= 13*11*9

= 1287

Therefore The number of different routes he can take is 1287.

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Answer:

x = 13

Step-by-step explanation:

Here, we want to get the missing side

from what we have, the triangle is a right angled triangle

In this kind of triangle, the Pythagoras’ theorem works

The square of the side facing the right angle ( marked x here) is equal to the sum of the squares of the two other sides

We have this as;

x^2 = 5^2 + 12^2

x^2 = 25 + 144

x^2 = 169

x = √169

x = 13

a) When Lily's age is 20, and Raphael's age is divided by 3 then the expression of Lily's age in terms of h is h < 10.

b) Ethan's age is half of Raphael's age therefore, Ethan's age is half Raphael's age.

a) If Raphael is twice as old as his sister Lily, we can represent their ages as 2h and h, respectively, where h is Lily's age. The sum of their ages is less than 30, so we can write the equation:

2h + h < 30

Combining like terms, we get:

3h < 30

Dividing both sides by 3, we get:

h < 10

This is an expression of Lily's age in terms of h.

b) If Ethan's age is 16, and Raphael's age is twice Ethan's age, we can represent Raphael's age as 2 × 16 = 32.

Ethan's age is 16, which is half of Raphael's age of 32. Therefore, Ethan's age is half of Raphael's age.

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The question is -

Raphael is twice as old as his sister. The sum of their ages is less than 30.

a) Lily's age is 20, and Raphael's age is divided by 3. Write an expression in terms of h for Lily's age.

b) An expression for Ethan's age 16. Write a sentence to describe how Ethan's age compares with Raphael's age.

The personnel director in this scenario is performing a convenience sampling.

Convenience sampling is a type of non-probability sampling method where the researcher selects participants based on their accessibility and proximity to the researcher. In this case, the personnel director chose employees from the department nearest her office for the study.

This approach may not provide a fully representative sample of the entire company's employee population, as it doesn't account for possible differences in eating habits across various departments. Therefore, the findings from this study might not accurately represent the proportion of employees who buy lunch in the cafeteria, bring their own lunches, or go out to lunch for the entire company.

To obtain more accurate results, the director might consider employing a probability sampling method, such as simple random sampling or stratified sampling, to ensure a more representative sample of the company's employees.

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The solution to the initial value problem y+y = 3+2cos(2r), y(0) = 0 is y(r) = 3/2 + cos(2r) - (3/2)cos(r). The behavior for large x tends towards a steady value

To solve the initial value problem, we can start by rewriting the equation as a first-order linear differential equation by introducing a new variable, v(r), such that v(r) = y(r) + y'(r).

Differentiating both sides of the equation with respect to r, we get v'(r) = 2cos(2r).

Integrating v'(r) with respect to r, we have v(r) = sin(2r) + C, where C is a constant.

Substituting y(r) + y'(r) back in for v(r), we have y(r) + y'(r) = sin(2r) + C.

To find C, we can use the initial condition y(0) = 0. Substituting r = 0 and y(0) = 0 into the equation, we get 0 + y'(0) = sin(0) + C, which gives us C = 0.

Therefore, the solution to the initial value problem is y(r) = 3/2 + cos(2r) - (3/2)cos(r).

Now, let's consider the behavior of the solution for large r (or x, since r and x are interchangeable in this context).

As r approaches infinity, the exponential term e^(-r) approaches zero. This means that the term Ce^(-r) becomes negligible compared to the other terms.

Therefore, the behavior of the solution for large x is primarily determined by the terms 3 + (1/2)sin(2r) - (1/4)cos(2r). The sin(2r) and cos(2r) terms oscillate between -1 and 1, but their coefficients (1/2 and -1/4, respectively) ensure that the amplitudes of the oscillations are limited.

Thus, for large x, the solution y approaches a steady value determined by the constant terms 3 - (1/4), which is approximately 2.75.

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The height at time t (in seconds) of a mass, oscillating at the end of a spring, is s(t) = 300 + 16 sin t cm. We have to find the velocity and acceleration at t = π/3 s.

Let's first find the velocity of the mass. The velocity of the mass is given by the derivative of the position of the mass with respect to time.t = π/3 s
s(t) = 300 + 16 sin t cm
Differentiating both sides of the above equation with respect to time
v(t) = s'(t) = 16 cos t cm/s

Now, let's substitute t = π/3 in the above equation,
v(π/3) = 16 cos (π/3) cm/s
v(π/3) = -8√3 cm/s

Now, let's find the acceleration of the mass. The acceleration of the mass is given by the derivative of the velocity of the mass with respect to time.t = π/3 s
v(t) = 16 cos t cm/s
Differentiating both sides of the above equation with respect to time
a(t) = v'(t) = -16 sin t cm/s²
Now, let's substitute t = π/3 in the above equation,
a(π/3) = -16 sin (π/3) cm/s²
a(π/3) = -8 cm/s²
Given, s(t) = 300 + 16 sin t cm, the height of the mass oscillating at the end of a spring. We need to find the velocity and acceleration of the mass at t = π/3 s.
Using the above concept, we can find the velocity and acceleration of the mass. Therefore, the velocity of the mass at t = π/3 s is v(π/3) = -8√3 cm/s, and the acceleration of the mass at t = π/3 s is a(π/3) = -8 cm/s².
At time t = π/3 s, the velocity of the mass is -8√3 cm/s, and the acceleration of the mass is -8 cm/s².

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The value of 6⅔ ÷ 12 is 5/9

The mathematical “operation” refers to calculating a value using operands and a math operator.

Mathematical operations include; Addition subtraction, multiplication, division. This operations are used to define the relationship between two terms.

PEDMAS is used for orderly arrangements for the operation.

6²/3 ÷ 12

We first convert the mixed fraction into improper fraction.

= 20/3 × 1/12

= 20/36

divide through by 4, then we have

= 5/9

Therefore the value of 6⅔ ÷ 12 is 5/9

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Nikita's distance from her for the second part of the race was 1.5 kilometers.

Since Nikita ran a 5-kilometer race in 39 minutes (0.65 of an hour) without training beforehand, and in the first part of the race, her average speed was 8.75 kilometers per hour, while for the second part of the race, she started to get tired, so her average speed dropped to 6 kilometers per hour, to determine which expression represents Nikita's distance for the second part of the race the following calculation must be performed:

Therefore, Nikita's distance from her for the second part of the race was 1.5 kilometers.

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Answer: The cost for the two pairs of shoes is $13.33

Step-by-step explanation:

Since we have given that

Cost of pair of flip flops = $3.46

Cost of pair of slip on shoes = $9.87

We need to find the cost of two pairs of shoes.

Cost of two pairs of shoes = Cost of pair of flip flops + Cost of pair of slip on shoes

\(cost=3.46+9.87=13.33\)

the cost for the two pairs of shoes is $13.33.

The margin of error for the 90% confidence interval is 0.012.

Given, a 90% confidence interval for the proportion of Americans with cancer was found to be (0.185, 0.210)

To calculate the margin of error, we can use the following formula:

margin of error = (upper limit of the confidence interval - lower limit of the confidence interval) / 2

Substitute the given values,

margin of error = (0.210 - 0.185) / 2 = 0.0125 ≈ 0.012

Therefore, the margin of error for the confidence interval (0.185, 0.210) is 0.012.

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9514 1404 393

Answer:

  ∆FAR ≅ ∆ORA

  HL theorem

Step-by-step explanation:

The only two overlapping triangles are ∆FAR and ∆ORA. The marks on their hypotenuses tell you those are the same length.

The common segment AR is a leg of each right triangle. Hence you have right triangles with a Hypotenuse and a Leg that are the same length. The HL theorem applies to tell you the triangles are congruent.

_____

Additional comment

Side FA is not identified as being congruent with anything, so SSS is not possible. The only angles marked as congruent are the right angles, so ASA and AAS are not possible. Because FA is not marked, there are no congruent angles between pairs of congruent sides, so SAS is not possible. The only congruence theorem it is possible to use here is HL.

Answer:

It's 70, big man

Step-by-step explanation:

It's a forth of 280

Answer:

D is correct answer (if its 22 and u accidentally wrote 24)

The numerical length of PQ is 20

A line segment in geometry is a section of a line that has two clearly defined endpoints and contains every point on the line that lies within its confines.

Given that the line segments are: -

OQ=11  O P = 9

The numerical value of PQ will be calculated as below.

OP+OQ=PQ

11+9=PQ=11+9=PQ

Therefore PQ=20=PQ

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Answer: Roberto is 26 years old and Tre is 12 years old.

Step-by-step explanation:

Let T=age of Tre

14+T=age of Roberto

14+T+T=28

14+2T=28

2T=14

T=7(Tre’s age 5 years ago)

14+T=14+7=21(Roberto’s age 5 years ago))

7+5=12 years old

21+5=26 years old

To check:

26-12=14

So Roberto is 14 years older.

a) The critical points of the function f are x = 8 and x = -9, which is option A. b) The function f is increasing on the open interval (-9, 8) and never decreasing i.e., option C and c) the function f may assume local maximum or minimum values at the endpoints x = -9 and x = 8.

a) To find the critical points of f, we need to find the values of x where the derivative f'(x) equals zero or is undefined. From the given derivative f'(x) = (x-8) ²(x+9), we can see that it is defined for all values of x. To find the critical points, we need to set f'(x) equal to zero and solve for x:

(x-8) ²(x+9) = 0

By setting each factor equal to zero, we can find the critical points:

x-8 = 0 or x+9 = 0

Solving these equations, we get:

x = 8 or x = -9

Therefore, the critical points of f are x = 8 and x = -9.

b) To determine where f is increasing or decreasing, we can examine the sign of the derivative f'(x) in different intervals. Considering the critical points x = 8 and x = -9, we can divide the number line into three intervals: (-∞, -9), (-9, 8), and (8, +∞).

c) Since f is increasing on the interval (-9, 8), it does not have any local maximum or minimum values within that interval. However, at the endpoints x = -9 and x = 8, f may assume local maximum or minimum values. To determine if these points correspond to local maximum or minimum, we need to examine the behavior of f around those points by evaluating f(x) itself.

Therefore, the answers to the questions are:

a) The critical points of f are x = 8 and x = -9. (Choice A).

b) The function is increasing on the open interval (-9, 8) and never decreasing. (Choice C).

c) The function f may assume local maximum or minimum values at x = -9 and x = 8, the endpoints of the interval.

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According to the statement the ratio of the corresponding sides of both triangles is equal.i.e., δWXY ~ δWVZ.

Given: δWXY is isosceles with legs WX and WY; δWVZ is isosceles with legs WV and WZ.To prove: δWXY ~ δWVZProof:In δWXY and δWVZ;WX = WY (Legs of isosceles triangle)WV = WZ (Legs of isosceles triangle)We have to prove δWXY ~ δWVZWe know that two triangles are similar when their corresponding sides are in the same ratio i.e., when they have the same shape.So, we have to prove that the ratio of the corresponding sides of both triangles is equal.(i) Corresponding sides WX and WVIn δWXY and δWVZ;WX/WV = WX/WZ (WZ is the corresponding side of WV)WX/WV = WY/WZ (WX is the corresponding side of WY)WX.WZ = WY.WV (Cross Multiplication).....(1)(ii) Corresponding sides WY and WZIn δWXY and δWVZ;WY/WZ = WX/WZ (WX is the corresponding side of WY)WY/WZ = WX/WV (WV is the corresponding side of WZ)WX.WZ = WY.WV (Cross Multiplication).....(2)From (1) and (2), we getWX.WZ = WY.WVHence, the ratio of the corresponding sides of both triangles is equal.i.e., δWXY ~ δWVZHence, Proved.

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To find the probability of drawing a 9 or a king from a well-shuffled deck of 52 cards, we need to determine the number of favorable outcomes and the total number of possible outcomes.

There are four kings in a deck (one king for each suit: hearts, diamonds, clubs, and spades), and there are four 9s in a deck (one 9 for each suit). However, one card (the king of hearts) is both a king and a 9. Therefore, we need to subtract this overlapping card from our count to avoid counting it twice.

The number of favorable outcomes (drawing a 9 or a king) is 4 (number of kings) + 4 (number of 9s) - 1 (overlapping card) = 7.

The total number of possible outcomes is 52 (since there are 52 cards in a deck).

Now, we can calculate the probability using the formula:

\(\[P(\text{{9 or king}}) = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of possible outcomes}}}}\]\)

\(\[P(\text{{9 or king}}) = \frac{7}{52}\]\)

This fraction cannot be further simplified since 7 and 52 do not share any common factors other than 1.

Therefore, the probability of drawing a 9 or a king from a well-shuffled deck of 52 cards is \(\( \frac{7}{52} \)\).

In LaTeX, the solution can be represented as:

\(\[P(\text{{9 or king}}) = \frac{7}{52}\]\)

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Answer:

Collect Like terms...Then Take the LCM

1/4x -1/3x +3 - 2

-1/12x + 1

Or

1 - 1/12x .

Option D.

Answer:

4.75x=54+2.5x    x=24

Step-by-step explanation:

x=Number of games

4.75x=54+2.5x

Subtract 2.5x from both sides:

2.25x=54

Divide both sides by 2.25:

x=24

Answer:

  AC = 10√3 cm

Step-by-step explanation:

The side lengths of a 30°-60°-90° triangle have the ratios 1 : √3 : 2.

In this geometry, ...

  BC = 2·BD = 2(5 cm) = 10 cm.

  AC = √3×BC = 10√3 cm

The values in the missing observation are:

X1    X2

76 22  

82 25

91 32

101    41

88 28

An expression contains terms with addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

x1 x2

76 22  

82 __

91 32

__    41

88 28

The difference between each row are:

76 - 22 = 54

82 - 25 = 57

91 - 32 = 59

101 - 41 = 60

88 - 28 = 60

We see that,

57 - 54 = 3

59 - 57 = 2

60 - 59 = 1

60 - 60 = 0

So,

The difference between the numbers is consecutive as 3, 2, 1, 0.

Thus,

The missing values of X1 and X2 are 101 and 25.

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f(6) = -4

given:

\(f(x) = -x+2\)

find:

\(f(6) \)

~~~~~~~~~~~~

➡️ \( f(6) = -(6) + 2\)

➡️ \( f(6) = -(6 -2) \)

➡️ \( f(6) = -4 \)

Answer:

"Simplifying rational expressions will make the further calculations easier since the variables to work with will usually be smaller.

To determine that a rational expression is in simplest form we need to make sure that the numerator and the denominator have no common variables."

Step-by-step explanation:

The rate of change of y with respect to x when t = 3 is = 3/25 and  the answer is not one of the options given.

An object moves in the xy-plane so that its position at any time tis given by the parametric equations x(t) = t° - 3t + 2 and y (t) = /t² + 16.

To find the rate of change of y with respect to x, we need to find dy/dx, which can be computed using the chain rule of differentiation:

(dy/dt) / (dx/dt) = dy/dx

We first compute dx/dt and dy/dt:

dx/dt = 1 - 3 = -2

dy/dt = 2t / (t² + 16)

When t = 3, we have:

dx/dt = -2

dy/dt = 2(3) / (3² + 16) = 6/25

Therefore, the rate of change of y with respect to x when t = 3 is:

(dy/dt) / (dx/dt) = (6/25) / (-2) = -3/25

So, the answer is not one of the options given.

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Answer: 7825 calories

Step-by-step explanation:

Feom the question, we are told that the meals have the following calories content meals having the following calorie counts: 975,520, 1560

.872, 1029,534,362,954, and 1019.

The sum of the calories will be:

= 975 + 520 +1560 + 872 + 1029 + 534 + 362 + 954 + 1019.

= 7825 calories

Given expression:  \(\frac{9x^2 - 1}{12x^2 - 12x} \div \frac{9x^2 + 12x + 3}{3x^2 - 6x + 3}\) . We can simplify the given expression by multiplying the numerator and denominator of the first fraction by (3x-1) and then factorise. So, the simplified expression is \($\frac{3(x-1)^2}{4x(3x+1)(x+1)}$\).

In the second fraction, we can factorise the quadratic expression in the numerator.

=  \(\frac{9x^2 - 1}{12x^2 - 12x} \cdot \frac{3x^2 - 6x + 3}{9x^2 + 12x + 3}\)

= \(\frac{(3x-1)(3x+1)}{12x(x-1)} \cdot \frac{3(x^2 - 2x + 1)}{3(3x^2 + 4x + 1)}\)

Simplify the expression.

= \(\frac{(3x-1)(x-1)}{4x(x-1)} \cdot \frac{(x-1)^2}{(3x+1)(x+1)}\)

= \(\frac{3(x-1)}{4x} \cdot \frac{(x-1)}{(3x+1)(x+1)}\)

= \(\frac{3(x-1)^2}{4x(3x+1)(x+1)}\) . Thus, the simplified expression is \($\frac{3(x-1)^2}{4x(3x+1)(x+1)}$\).

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