If a, b and c are distinct real numbers, prove that the equation(x−a)(x−b)+(x−b)(x−c)+(x−c)(x−a=0has real and distinct roots.

Answers

Answer 1

Answer:

mail

Step-by-step explanation:

skating Dinero broke 1p revision yahoo d10


Related Questions

Quadrilateral ABCD is a parallelogram. Segment BD is a diagonal of the parallelogram.

Which statement and reason correctly complete this proof?

Answers

Answer:

  (A)  alternate interior angles

Step-by-step explanation:

You want the missing statement in the proof that opposite angles of a parallelogram are congruent.

Proof

The proof here shows angles A and C are congruent because they are corresponding parts of congruent triangles. To get there, the triangles must be shown to be congruent.

In statement 5, the triangles area claimed congruent by the ASA theorem, which requires two corresponding pairs of angles and congruent sides.

In statement 4, the relevant sides are shown congruent, so it is left to statement 3 to show two pairs of angles are congruent.

Of the offered answer choices, only one of them deals with two pairs of angles. Answer choice A is the correct one.

7. The Key West Lighthouse is 86 feet tall. What is the height of the lighthouse in meters?

Answers

The height of the Key West Lighthouse in meters is approximately 26.21 meters.

Here's how you can calculate it:

- There are 3.28 feet in a meter.

- Divide the height of the lighthouse in feet by the number of feet in a meter: 86 ÷ 3.28 = 26.21 meters (rounded to two decimal places).

- Therefore, the height of the Key West Lighthouse in meters is approximately 26.21 meters.

A spring with a mass of 2 kg has damping constant 10, and a force of 4 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1 m beyond its natural length and then released with zero velocity. Find the position (in m) of the mass at any time t. Xm 6

Answers

The position of the mass of the object 2kg at time t =1s is equal to -3.97m approximately .

Mass of the object 'm' = 2 kg

Damping constant 'c' = 10

Spring constant 'k' = F/x

                               = 4 N / 0.5 m

                               = 8 N/m

F(t) is any external force applied to the object

x is the displacement of the object from its equilibrium position

x(0) = 1 m (initial displacement)

x'(0) = 0 (initial velocity)

Equation of motion for a spring-mass system with damping is,

mx'' + cx' + kx = F(t)

Substituting these values into the equation of motion,

Since there is no external force applied

2x'' + 10x' + 8x = 0

This is a second-order homogeneous differential equation with constant coefficients.

The characteristic equation is,

2r^2 + 10r + 8 = 0

Solving for r, we get,

⇒ r = (-10 ± √(10^2 - 4× 2× 8)) / (2×2)

     =( -10 ± 6 )/ 4

     = ( -2.5 ± 1.5 )

The general solution for x(t) is,

x(t) = e^(-5t) (c₁ cos(t) + c₂ sin(t))

Using the initial conditions x(0) = 1 and x'(0) = 0, we can solve for the constants c₁ and c₂

x(0) = c₁

      = 1

x'(t) = -5e^(-5t) (c₁ cos(t) + c₂ sin(t)) + e^(-5t) (-c₁ sin(t) + c₂ cos(t))

x'(0) = -5c₁ + c₂ = 0

 ⇒-5c₁ + c₂ = 0

⇒ c₂ = 5c₁  = 5

The solution for x(t) is,

x(t) = e^(-5t) (cos(t) + 5 sin(t))

The position of the mass at any time t is given by x(t),

Plug in any value of t to find the position.

For example, at t = 1 s,

x(1) = e^(-5) (cos(1) + 5 sin(1))

     ≈ -3.97 m

The position of the mass oscillates sinusoidally and decays exponentially due to the damping.

Therefore, the position of the mass at t = 1 s is approximately -3.97 m.

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What is the domain and range of g(x)=-|x|

Answers

Answer:

Step-by-step explanation

Domain :

x

>

4

, in interval notation :

(

4

,

)

Range:

g

(

x

)

R

, in interval notation :

(

,

)

Explanation:

g

(

x

)

=

ln

(

x

4

)

;

(

x

4

)

>

0

or

x

>

4

Domain :

x

>

4

, in interval notation :

(

4

,

)

Range: Output may be any real number.

Range:

g

(

x

)

R

, in interval notation :

(

,

)

graph{ln(x-4) [-20, 20, -10, 10]} [Ans]  x>4

Answer:

Step-by-step explanation:

The Domain of g(x) = -|x| is all real numbers (no restrictions on what values x can take).

The Range of g(x) = -|x| is all real numbers less than or equal to zero. Absolute value of any real number is always greater than or equal to zero, and multiplying by a negative sign, that flips the sign of the result. So, g(x) will always be less than or equal to zero.

Domain:  (-∞, ∞), {x|x ∈ R}

Range: (-∞, 0), {y ≤ 0}

Use undetermined coefficients to find the particular solution to
y' +41 -53 = - 580 sin(2t)
Y(t) = ______

Answers

To find the particular solution to this differential equation using undetermined coefficients, we first need to guess the form of the particular solution. Since the right-hand side of the equation is a sinusoidal function, our guess will be a linear combination of sine and cosine functions with the same frequency:

y_p(t) = A sin(2t) + B cos(2t)

We can then find the derivatives of this guess:

y'_p(t) = 2A cos(2t) - 2B sin(2t)
y''_p(t) = -4A sin(2t) - 4B cos(2t)

Substituting these into the differential equation, we get:

(-4A sin(2t) - 4B cos(2t)) + 41(2A cos(2t) - 2B sin(2t)) - 53(A sin(2t) + B cos(2t)) = -580 sin(2t)

Simplifying and collecting terms, we get:

(-53A + 82B) cos(2t) + (82A + 53B) sin(2t) = -580 sin(2t)

Since the left-hand side and right-hand side of this equation must be equal for all values of t, we can equate the coefficients of each trigonometric function separately:

-53A + 82B = 0
82A + 53B = -580

Solving these equations simultaneously, we get:

A = -23
B = -15

Therefore, the particular solution to the differential equation is:

y_p(t) = -23 sin(2t) - 15 cos(2t)

Adding this to the complementary solution (which is just a constant, since the characteristic equation has no roots), we get the general solution:

y(t) = C - 23 sin(2t) - 15 cos(2t)

where C is a constant determined by the initial conditions.
To solve the given differential equation using the method of undetermined coefficients, we need to identify the correct form of the particular solution.

Given the differential equation:
y'(t) + 41y(t) - 53 = -580sin(2t)

We can rewrite it as:
y'(t) + 41y(t) = 53 + 580sin(2t)

Now, let's assume the particular solution Y_p(t) has the form:
Y_p(t) = A + Bsin(2t) + Ccos(2t)

To find A, B, and C, we will differentiate Y_p(t) with respect to t and substitute it back into the differential equation.

Differentiating Y_p(t):
Y_p'(t) = 0 + 2Bcos(2t) - 2Csin(2t)

Now, substitute Y_p'(t) and Y_p(t) into the given differential equation:
(2Bcos(2t) - 2Csin(2t)) + 41(A + Bsin(2t) + Ccos(2t)) = 53 + 580sin(2t)

Now we can match the coefficients of the similar terms:
41A = 53 (constant term)
41B = 580 (sin(2t) term)
-41C = 0 (cos(2t) term)

Solving for A, B, and C:
A = 53/41
B = 580/41
C = 0

Therefore, the particular solution is:
Y_p(t) = 53/41 + (580/41)sin(2t)

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Let ∑an be a convergent series, and let S=limsn, where sn is the nth partial sum

Answers

The given statement "If ∑an is a convergent series, then S = limsn, where sn is the nth partial sum. " is true. This is because the sum of the series is defined as the limit of the sequence of partial sums.

Given that ∑an is a convergent series, sn is the nth partial sum, S=limsn

To prove limn→∞ an = 0

Since ∑an is convergent, we know that the sequence {an} must be a null sequence, i.e., it converges to 0. This means that for any ε>0, there exists an N such that |an|<ε for all n≥N.

Now, let's consider the partial sums sn. We know that S=limsn, which means that for any ε>0, there exists an N such that |sn−S|<ε for all n≥N.

Using the triangle inequality, we can write:

|an|=|sn−sn−1|≤|sn−S|+|sn−1−S|<2ε

Therefore, we have shown that limn→∞ |an| = 0, which implies limn→∞ an = 0, as required.

Hence, the proof is complete.

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Ken bought a car last year to drive back and forth to work. Last year he spent $1,098 on gas. This year, it was $1,562. What is the inflation rate?​

Answers

The inflation rate for Ken's gas expenses between the two years is approximately 42.26%.

To calculate the inflation rate for Ken's gas expenses, we can use the following formula: (Current Year Expense - Previous Year Expense) / Previous Year Expense × 100%.

In this case, the previous year's gas expense was $1,098 and the current year's expense is $1,562.

To find the difference in expenses, subtract the previous year's expense from the current year's expense: $1,562 - $1,098 = $464.

Now, divide this difference by the previous year's expense: $464 / $1,098 ≈ 0.4226.

Finally, multiply the result by 100% to get the inflation rate: 0.4226 × 100% ≈ 42.26%.

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at a party, seven gentlemen check their hats. in how many ways can their hats be returned so that 1. no gentleman receives his own hat? 2. at least one of the gentlemen receives his own hat? 3. at least two of the gentlemen receive their own hats?

Answers

1) There are 1854 ways to return the hats so that no gentleman receives his own hat.

2) There are 3186 ways to return the hats so that at least one of the gentlemen receives his own hat.

3) There are 865 ways to return the hats so that at least two of the gentlemen receive their own hats.

1) This problem involves the concept of permutations. A permutation is an arrangement of objects in a particular order. In this case, we need to find the number of permutations for returning the hats of the gentlemen.

To find the number of ways that no gentleman receives his own hat, we can use the principle of derangements. A derangement is a permutation of a set of objects such that no object appears in its original position.

The number of derangements of a set of n objects is denoted by !n and can be calculated using the formula:

!n = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)

For n = 7, we have

!7 = 7!(1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5! + 1/6!)

= 1854

Therefore, there are 1854 ways to return the hats so that no gentleman receives his own hat.

2) To find the number of ways that at least one of the gentlemen receives his own hat, we can use the complementary principle. The complementary principle states that the number of outcomes that satisfy a condition is equal to the total number of outcomes minus the number of outcomes that do not satisfy the condition.

The total number of ways to return the hats is 7!, which is 5040. The number of ways that no gentleman receives his own hat is 1854 (as we found in part 1). Therefore, the number of ways that at least one of the gentlemen receives his own hat is

5040 - 1854 = 3186

Therefore, there are 3186 ways to return the hats so that at least one of the gentlemen receives his own hat.

3) To find the number of ways that at least two of the gentlemen receive their own hats, we can use the inclusion-exclusion principle. The inclusion-exclusion principle states that the number of outcomes that satisfy at least one of several conditions is equal to the sum of the number of outcomes that satisfy each condition minus the sum of the number of outcomes that satisfy each pair of conditions, plus the number of outcomes that satisfy all of the conditions.

In this case, the conditions are that each of the seven gentlemen receives his own hat. The number of outcomes that satisfy each condition is 6!, which is 720. The number of outcomes that satisfy each pair of conditions is 5!, which is 120. The number of outcomes that satisfy all of the conditions is 4!, which is 24.

Using the inclusion-exclusion principle, the number of outcomes that satisfy at least two of the conditions is

6! - (7C₂)5! + (7C₃)4! - (7C₄)3! + (7C₅)2! - (7C₆)1! + 0!

= 720 - (21)(120) + (35)(24) - (35)(6) + (21)(2) - (7)(1) + 0

= 720 - 2520 + 840 - 210 + 42 - 7 + 0

= 865

Therefore, there are 865 ways to return the hats so that at least two of the gentlemen receive their own hats.

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What is the area of the trapezoid?​

Answers

Answer:

33

Step-by-step explanation:

Pythagorean theorem:

6,5^ - 2.5^2= 36

✓36=6 second leg

3×6=18 square area

0,5×6×2,5=7,5 area of a triangle

2×7,5 + 18= 33

Pls help

label each scatterplot correctly,
no association
linear negative association linear positive association
nonlinear association

Answers

Without a specific set of scatterplots to examine, I can provide some general guidelines for labeling scatterplots based on their association:

1. No association: When there is no pattern or relationship between the two variables being plotted, we label the scatterplot as having no association.

2. Linear positive association: When the points in the scatterplot form a roughly straight line that slopes upwards from left to right, we label the scatterplot as having a linear positive association. This means that as the value of one variable increases, the value of the other variable also tends to increase.

3. Linear negative association: When the points in the scatterplot form a roughly straight line that slopes downwards from left to right, we label the scatterplot as having a linear negative association. This means that as the value of one variable increases, the value of the other variable tends to decrease.

4. Nonlinear association: When the points in the scatterplot do not form a straight line, we label the scatterplot as having a nonlinear association. This means that the relationship between the two variables is more complex and cannot be described simply as a straight line. There are many different types of nonlinear relationships, including curves, U-shaped or inverted-U-shaped patterns, and more.

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54/g - h when g=6 and h=3

Answers

Answer:

18

Step-by-step explanation:

54/g - h                              g = 6 and h = 3

54/6 - 3

= 54/3

= 18

So, the answer is 18.

find the equation of the line that has a gradient of 2 and passes through the point (-3,3)​

Answers

Answer:

[tex]y = 2x + 9[/tex]

Step-by-step explanation:

It is given that the slope of the line is 2, and it passes through (-3 , 3). The equation of straight lines is y = mx + b, in which:

y = (x , y) = 3

m = slope (gradient) = 2

x = (x , y) = -3

b = y-intercept

~

Plug in the corresponding numbers to the corresponding variables:

y = mx + b

3 = (2)(-3) + b

First, multiply -3 with 2:

[tex]3 = (2)(-3) + b\\3 = (2 * -3) + b\\3 = -6 + b[/tex]

Next, isolate the variable, b. Note the equal sign, what you do to one side, you do to the other. Add 6 to both sides of the equation:

[tex]3 = b - 6\\3 (+6) = b - 6 (+6)\\b = 3 + 6\\b = 9[/tex]

Plug in 2 for slope, and 9 for y-intercept, in the given equation:

[tex]y = mx + b\\m = 2\\b = 9\\[/tex]

[tex]y = 2x + 9[/tex] is your answer.

~

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At midnight, the temperature in a city was 5 degrees celsius. the temperature was dropping at a steady rate of 1 degrees celsius per hour.

a. write an inequality that represents t, the number of hours past midnight, when the temperature was colder than -3 degrees celsius.

b. explain or show your reasoning.

Answers

The inequality that represents t, the number of hours past midnight, when the temperature was colder than -3 degrees Celsius is t > 8.

The inequality that represents t, the number of hours past midnight, when the temperature was colder than -3 degrees Celsius is t > 8.

When the temperature drops at a steady rate of 1 degree Celsius per hour, it will take 8 hours to reach -3 degrees Celsius from the initial temperature of 5 degrees Celsius.

Therefore, any time past 8 hours after midnight will result in a temperature colder than -3 degrees Celsius.

Thus, the inequality t > 8 represents the number of hours past midnight when the temperature was colder than -3 degrees Celsius.

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Construct the class boundaries for the following frequency distribution table. also construct less than cumulative and greater than cumulative frequency tables.
ages:- 1 - 3, 4-6, 7-9, 10-12, 13-15
no of children:- 10,12,15,13,9​

Answers

The class boundaries are 0.5 - 3.5, 3.5 - 6.5, 6.5 - 9.5, 9.5 - 12.5, 12.5 - 15.5.

To find the class boundaries, we need to add and subtract 0.5 from the upper and lower limits of each class interval, respectively.

Using this formula, we get the following class boundaries:

Class Boundaries:

0.5 - 3.5, 3.5 - 6.5, 6.5 - 9.5, 9.5 - 12.5, 12.5 - 15.5

To construct the less than cumulative frequency table, we need to add up the frequencies of all the classes up to each class. For example:

Less than Cumulative Frequency Table:

Ages No. of Children Cumulative Frequency

1-3 10 10

4-6 12 22

7-9 15 37

10-12 13 50

13-15 9 59

To construct the greater than cumulative frequency table, we need to subtract the frequency of each class from the total frequency and then add the resulting values up to obtain the cumulative frequency. For example:

Greater than Cumulative Frequency Table:

Ages No. of Children Cumulative Frequency

13-15 9 59

10-12 13 50

7-9 15 37

4-6 12 22

1-3 10 10

Note that the last value of the greater than cumulative frequency table is always equal to the total frequency, which in this case is 59.

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Which shape contains two pairs of parallel lines? A. shape A B. shape B C. shape C D. shape D

Answers

Answer: C

Step-by-step explanation:

C is a parallelogram, meaning that both sets of opposite sides are parallel.

2. Hamilton claimed that there are only 4 circuits that begin with the letters LTSR Q. Find them. 3. Find all four possible Hamiltonian circuits that begin with JVTSR

Answers

To find the possible Hamiltonian circuits that begin with JVTSR, we can start by constructing a path that begins with JVTSR and visits each vertex exactly once. Such a path must be of the form JVTSRX, where X is the remaining vertex.

Case 1: JVTSRQX

To find the possible value of X, we note that the only edges incident to J are V and T, and the only edges incident to Q are S and R. Thus, we must have X = S or X = R, and the circuits are JVTSRQS and JVTSRQR.

Case 2: JVTSRXQ

To find the possible value of X, we note that the only edges incident to X are S and L. Thus, we must have X = L, and the circuit is JVTSRLQ.

Case 3: JVTSRLX

To find the possible value of X, we note that the only edges incident to J are V and T, and the only edges incident to L are T and R. Thus, we must have X = R, and the circuit is JVTSRLR.

Case 4: JVTSRXL

To find the possible value of X, we note that the only edges incident to Q are S and R, and the only edges incident to X are L and S. Thus, we must have X = L, and the circuit is JVTSRQL.

Therefore, there are four possible Hamiltonian circuits that begin with JVTSR: JVTSRQS, JVTSRQR, JVTSRLQ, and JVTSRLR.

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using graphical method to solve simultaneous equation y=2-2x and y=2x-6

Answers

The solution to the system of equations is x=2 and y=-2.

To solve the system of simultaneous equations graphically, we need to graph both equations on the same coordinate plane and find their point of intersection.

First, we'll rearrange both equations to be in the form y=mx+b, where m is the slope and b is the y-intercept.

y = 2 - 2x can be rewritten as y = -2x + 2

y = 2x - 6 can be rewritten as y = 2x - 6

Now, we'll plot both equations on the same coordinate plane. To do this, we'll create a table of values for each equation and plot the points.

For y = -2x + 2: (0,2), (1,0), (2,-2)

For y = 2x - 6:(0,-6), (1,-4), (2,-2)

Next, we'll plot these points on the same graph and draw the lines connecting them.

The point where the lines intersect is the solution to the system of equations. From the graph, we can see that the point of intersection is (2,-2).

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4. The perimeter of an isosceles trapezoid ABCD is 27. 4 inches. If BC = 2 (AB), find AD, AB, BC, and CD. ​

Answers

The lengths of the sides are: AB = CD = 4.5667 inches; BC = 9.1333 inches and AD = 9.1333 inches

An isosceles trapezoid is a four-sided figure with two parallel sides and two non-parallel sides that are equal in length. In this problem, we are given that the perimeter of the isosceles trapezoid ABCD is 27.4 inches, and that BC is twice as long as AB.

Let's start by assigning variables to the lengths of the sides. Let AB = x, BC = 2x, CD = x, and AD = y. Since the perimeter of the trapezoid is the sum of all four sides, we can write the equation:

x + 2x + x + y = 27.4

Simplifying the equation, we get:

4x + y = 27.4

We also know that the non-parallel sides of an isosceles trapezoid are equal in length, so we can write:

AB = CD = x

Now we can use the fact that BC is twice as long as AB to write:

BC = 2AB

Substituting x for AB, we get:

2x = BC

Now we can use the Pythagorean theorem to find the length of AD. The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs (the shorter sides) is equal to the square of the hypotenuse (the longest side). Since AD is the hypotenuse of a right triangle, we can write:

AD^2 = BC^2 - (AB - CD)^2

Substituting the values we know, we get:

y^2 = (2x)^2 - (x - x)^2

Simplifying, we get:

y^2 = 4x^2

Taking the square root of both sides, we get:

y = 2x

Now we can use the equation we found earlier to solve for x:

4x + y = 27.4

4x + 2x = 27.4

6x = 27.4

x = 4.5667

Now we can find the lengths of the other sides:

AB = CD = x = 4.5667

BC = 2AB = 2x = 9.1333

AD = y = 2x = 9.1333

So the lengths of the sides are:

AB = CD = 4.5667 inches

BC = 9.1333 inches

AD = 9.1333 inches

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in a psychology class, 37 students have a mean score of 86.9 on a test. then 22 more students take the test and their mean score is 74.4. what is the mean score of all of these students together? round to one decimal place.

Answers

The mean score of all the students together is 83.1 (rounded to one decimal place).

The mean score of all the students together can be calculated using the formula:

(mean score of first group * number of students in first group + mean score of second group * number of students in second group) / (total number of students)

Substituting the values, we get:

(86.9 * 37 + 74.4 * 22) / (37 + 22)

= (3215.3 + 1636.8) / 59

= 4852.1 / 59

= 82.3

Therefore, the mean score of all the students together is 82.3, rounded to one decimal place.

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Michael and Susan are a combined height of 132 inches. If Michael is 71
inches tall, how tall is Susan?

Answers

Answer: 61 in.

Step-by-step explanation:

What you do first is you must find the total number if inches of both humans combined

132 in.

Then, you want to take the 71 in. from Michael's height, and subtract it from the total number.

132

-71

61

----------

61 in. is your answer.

Niamh was driving back home following a business trip.
She looked at her Sat Nav at 17:30
Time: 17:30
Distance: 143 miles
Niamh arrived home at 19:42
Work out the average speed of the car, in mph, from 17:30 to 19:42
You need to show all your working
:)​

Answers

Answer:

65 mph

Step-by-step explanation:
To calculate the average speed of Niamh's car, we need to use the formula:

Average speed = Total distance ÷ Total time

First, we need to calculate the total time elapsed from 17:30 to 19:42:

Total time = 19:42 - 17:30 = 2 hours and 12 minutes

To convert the minutes to decimal form, we divide by 60:

2 hours and 12 minutes = 2 + (12 ÷ 60) = 2.2 hours

Now we can calculate the average speed:

Average speed = Total distance ÷ Total time

Average speed = 143 miles ÷ 2.2 hours

Average speed = 65 mph

Therefore, the average speed of Niamh's car from 17:30 to 19:42 was 65 mph.

I MAKE U BRAINLIEST solve for x

Answers

Answer:  9

Step-by-step explanation:

The angle is 1/2 of the arc angle

Since the tangent line is a line, I know the angle on the other side of 78 is

180-78 = 102

That angle, 102, is 1/2 the arc angle

102  = 1/2 (23x -3)                 > multiply both sides by 2

204 =  23x -3                        > add 3 to both sides

207 = 23x                             >divide both sides by 23

x=9

Answer is 9

Source trust me bro

EASY POINTS!!
i need someone to write three sentences that explains how i got the answer i have the equation already but dont know how to do it THANKS SO MUCH.

An amusement park has discovered that the brace that provides stability to the Ferris wheel has been damaged and needs work. The arc length of the steel reinforcement that must be replaced is between the two seats shown below. The sector area is 28.25 ft2 and the radius is 12 feet. What is the length of steel that must be replaced (Arc Length)? Describe the steps you used to find your answer and show all work. Round θ to the nearest tenth.
my "work":
Area of Sector = 28.25 ft² & Radius = 12 feet

Area of sector = ∅/360 × π × r²

Put the values,

28.25 = ∅/360 × π × 12²

∅ = (28.25 × 360) / π×12²

∅ = 22.47 ≈ 22.5

length of arc =∅/360 × 2 × π × r

L = 22.5/360 × 2 × π × 12

L = 4.71 Feet

Answers

We used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation

The Explanation of your solution

First, we used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation and solving for ∅, which resulted in ∅ ≈ 22.5 degrees.

Next, we applied the arc length formula, L = ∅/360 × 2 × π × r, and plugged in the values we had, including the calculated ∅ and the given radius (12 feet).

Finally, we calculated the arc length (L) to be approximately 4.71 feet, which is the length of steel that must be replaced.

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when rounding to the nearest hundred what is the greatest whole number that rounds to 500?

Answers

Answer:

499

Step-by-step explanation:

Consider right angle triangle ABC, right angled at B. If AC=17 units and BC+8 units determine all the trigonometric ratios of angle C

Answers

The trigonometric ratios of angle C are sin C = 15/17, cos C = 8/17, and tan C = 15/8.

Since triangle ABC is a right triangle with a right angle at B, and we know AC = 17 units (hypotenuse) and BC = 8 units (adjacent side to angle C), we can use the Pythagorean theorem to find the length of the remaining side, AB (opposite side to angle C).

The Pythagorean theorem states: AB² + BC² = AC²

Plugging in the values we know:
AB² + 8² = 17²
AB² + 64 = 289

To find AB:
AB² = 289 - 64 = 225
AB = √225 = 15 units

Now we can determine the trigonometric ratios of angle C:

1. sine (sin C) = opposite/hypotenuse = AB/AC = 15/17
2. cosine (cos C) = adjacent/hypotenuse = BC/AC = 8/17
3. tangent (tan C) = opposite/adjacent = AB/BC = 15/8

So the trigonometric ratios of angle C are:
sin C = 15/17, cos C = 8/17, and tan C = 15/8.

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What is the x intercept of f(x)= 2x^2+5x+3

Answers

Answer: x intercepts = (-1.5,0) and (-1,0)

Step-by-step explanation: Graphed it in desmos :)

Worth 50 points!!! a ball is dropped from a height of 32 meters. with each bounce, the ball reaches a height that is half the height of the previous bounce. after which bounce will the ball rebound to a maximum height of 25 centimeters?

Answers

The ball will rebound to maximum height of 25 centimetres or 0.25 meters after 7 bounces.

Firstly perform the unit conversion. As known, 1 meter is 100 cm. So, 25 centimetres is 0.25 meters.

Now, the formula to be used to find the number of bounces is -

New height × [tex] {2}^{n} [/tex] = old height, where n refers to number of bounces.

Keeping the values in formula

0.25 × [tex] {2}^{n} [/tex] = 32

Rearranging the equation

[tex] {2}^{n} [/tex] = 32/0.25

Divide the values

[tex] {2}^{n} [/tex] = 128

Converting the result into exponent form

[tex] {2}^{n} [/tex] = 2⁷

Thus, n will be 7 bounces.

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A rectangular pyramid fits exactly on top of a rectangular prism. The prism* 1 point has a length of 26 cm, a width of 5 cm, and a height of 14 cm. The pyramid has a height of 23 cm. Find the volume of the composite space figure. Round to the nearest hundredth .

Answers

The volume of the composite space figure is approximately 2818.33 cubic cm.

How to calculate the  volume of the composite space figure

To find the volume of the composite space figure, we need to add the volumes of the rectangular prism and the rectangular pyramid.

The rectangular prism has a length of 26 cm, a width of 5 cm, and a height of 14 cm. So its volume is:

V_prism = length x width x height

V_prism = 26 cm x 5 cm x 14 cm

V_prism = 1820 cubic cm

The rectangular pyramid has a height of 23 cm and a rectangular base with a length of 26 cm and a width of 5 cm. To find its volume, we need to first find its base area:

A_base = length x width

A_base = 26 cm x 5 cm

A_base = 130 square cm

Then, we can use the formula for the volume of a pyramid:

V_pyramid = (1/3) x base area x height

V_pyramid = (1/3) x 130 square cm x 23 cm

V_pyramid = 998.33 cubic cm (rounded to the nearest hundredth)

To find the total volume of the composite space figure, we add the volumes of the prism and the pyramid:

V_total = V_prism + V_pyramid

V_total = 1820 cubic cm + 998.33 cubic cm

V_total = 2818.33 cubic cm (rounded to the nearest hundredth)

Therefore, the volume of the composite space figure is approximately 2818.33 cubic cm.

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The altitude to the hypotenuse of a right angled triangle is 8 cm. If the hypotenuse is 20 cm long, find the lenghs of the two segments of the hypotenuse

Answers

Let the two segments of the hypotenuse be x and y.
Using the Pythagorean theorem, we know that:
x^2 + 8^2 = y^2
and
y^2 + 8^2 = 20^2
Simplifying the second equation:
y^2 = 20^2 - 8^2
y^2 = 336
y = sqrt(336) = 4sqrt(21)
Now we can use the first equation to solve for x:
x^2 + 8^2 = (4sqrt(21))^2
x^2 + 64 = 336
x^2 = 272
x = sqrt(272) = 4sqrt(17)
Therefore, the lengths of the two segments of the hypotenuse are 4sqrt(17) cm and 4sqrt(21) cm.

Find the absolute maximum and absolute minimum values off on each interval. (If an answer does not exist, enter DNE.) F(x) = 2x² - 16x + 850 (a) (0,4) Absolute maximum: Absolute minimum: (b) (0,4) Absolute maximum: Absolute minimum:

Answers

From the above information we get:

Absolute maximum: 850
Absolute minimum: 800

To find the absolute maximum and minimum values of the function f(x) = 2x² - 16x + 850 on the given interval (0,4), we will follow these steps:

1. Find the critical points by taking the first derivative of f(x) and setting it equal to zero.
2. Determine if the critical points are within the interval (0,4).
3. Evaluate f(x) at the critical points and endpoints of the interval.
4. Identify the absolute maximum and minimum values based on the results.

Step 1: Find the critical points
f'(x) = 4x - 16
Setting f'(x) equal to zero:
4x - 16 = 0
4x = 16
x = 4

Step 2: Determine if the critical point is within the interval (0,4)
The critical point x = 4 is within the interval (0,4).

Step 3: Evaluate f(x) at the critical points and endpoints of the interval
f(0) = 2(0)² - 16(0) + 850 = 850
f(4) = 2(4)² - 16(4) + 850 = 850 - 64 + 850 = 800

Step 4: Identify the absolute maximum and minimum values based on the results
Absolute maximum: f(0) = 850
Absolute minimum: f(4) = 800

To answer the question:
(a) Interval (0,4)
Absolute maximum: 850
Absolute minimum: 800

(b) It seems you have repeated the interval (0,4), so the answer remains the same.
Absolute maximum: 850
Absolute minimum: 800

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