Express the lengths a and b in the figure in terms of θ.

Answer:

Step-by-step explanation:

The end behavior of the rational polynomial function [tex]f(x) = \frac{6x^3 - x^2 + 7}{2x + 5}[/tex] is,

{x → ∞, y → ∞ and x → - ∞, y → ∞}.

A polynomial function's final behavior is how its graph behaves as x gets closer to positive or negative infinity.

The graph's final behavior is determined by a polynomial function's degree and leading coefficient.

Given, A rational polynomial function [tex]f(x) = \frac{6x^3 - x^2 + 7}{2x + 5}[/tex].

Now A cubic function divided by a linear function would result in a quadratic function, And as the coefficients of the highest degree terms of both the highest terms are positive the coefficient of the highest term of the quadratic function will also be positive and it's graph will be a parabola that opens upwards and symmetric about the y-axis.

Therefore, The end behavior will be when x tends to positive infinity y goes to positive infinity and when x tends to negative infinity y goes to positive infinity.

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The required measure of the angles x and y is given as 50°.

Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.

Here,

As it can be seen, x and y are alternate-opposite equal angles,
x = y

Now,
Adding complementary angles to 180°,
x + 60 + 70 = 180
x = 180 - 130
x = 50°

So,
x = y = 50°

Thus, the required measure of the angles x and y is given as 50°.

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Answer:[tex]\sqrt{5x[/tex]

Step-by-step explanation:

The radical equation solution is x= -1.

What is radical equation?

The unknown is a component of the radicand of a radical expression in a radical equation. Equations with an uncertain value enclosed by a radical sign are referred to as radical equations (also known as irrational) equations. Radical expressions are those that fall inside the square root. Radical inequality refers to an inequality contained within a radical.

Here the given radical equation is

=> [tex]\sqrt{2x+3} -\sqrt{x+2}[/tex] = 0

=> [tex]\sqrt{2x+3} = \sqrt{x+2}[/tex]

Now taking square on both sides then,

=> [tex](\sqrt{2x+3})^2 = (\sqrt{x+2})^2[/tex]

=> 2x+3 = x+2

=> 2x-x = 2-3

=> x = -1

Hence value of the x is -1.

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Answer:

5p

6d6errryd6tj4wytejytjwyreyrj

Answer:

A = - 12 , B = - 8 , C = - 4

Step-by-step explanation:

substitute the value of x above the letters into the equation and solve for y

x = - 2

2(- 2) - y = 8

- 4 - y = 8 ( add 4 to both sides )

- y = 12 ( multiply both sides by - 1 )

y = - 12 ← value of A

x = 0

2(0) - y = 8

0 - y = 8- y = 8 ( multiply both sides by - 1 )

y = - 8 ← value of B

x = 2

2(2) - y = 8

4 - y = 8 ( subtract 4 from both sides )

- y = 4 ( multiply both sides by - 1 )

y = - 4 ← value of C

Julian's average speed for the whole journey was 8 miles per hour.

To find Julian's average speed for the whole journey, we can use the formula:

average speed = total distance / total time

We know that Julian walked from his house to the library for 2 hours at a speed of 4 miles per hour and then returned home by bicycle at a speed of 12 miles per hour.

To find the total distance, we can add the distance he walked to the distance he rode his bicycle:

total distance = distance walked + distance rode by bicycle

To find the distance he walked, we can use the formula:

distance = speed x time

distance walked = 4 miles/hour x 2 hours = 8 miles

distance rode by bicycle = 12 miles/hour x 2 hours = 24 miles

total distance = 8 miles + 24 miles = 32 miles

We know that Julian walked for 2 hours and rode for 2 hours, so we can add those times to find the total time:

total time = time walked + time rode by bicycle = 2 hours + 2 hours = 4 hours

Now we can use the formula to find Julian's average speed for the whole journey:

average speed = total distance / total time = 32 miles / 4 hours = 8 miles/hour

Therefore, Julian's average speed for the whole journey was 8 miles per hour.

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The graph of the piece-wise function f(x) is given by the image presented at the end of the answer.

A piece-wise function is a function that has different definitions, depending on the input of the function.

The extreme points of the function in this problem are given as follows:

Hence the graph of the function in this problem is composed as follows:

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Answer:

X = 5

Step-by-step explanation:

this must form a triangle so simply using the same value of x equally in each equation.

The distance, y, in miles, covered by a car during a specific period, x, in hours and The distance, y, in miles, a car travels in a given period of time, x, in hours

The following graph displays the distance travelled as a linear function of time, D (in miles), and (in hours): To view a larger version of the graph, click on it.) Find AD and 4 values between the specified start and finish times for each interval: Fill in your responses in the appropriate columns in the table below: AD t =1tot=3.5, 0.5 to t =3.5, t=2tot=4, and AD t =3tot=4

b) In light of your findings in (a), it is evident that the average rate of change of D is constant. Decide on ALL that apply (more than one may apply) A It symbolises the car's velocity: B. The angle of the line's slope: C. The average speed of the car throughout the first two hours.

Therefore the correct answer is option A ) It travels for 3 hours, then stops for 2 hours, and finally travels again for 2 hours.

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The required point that is not always true for parallelograms is the diagonals are perpendicular.

A parallelogram is a quadrilateral consisting of pairs of parallel sides.

Here,
For any parallelogram, the point that remains always the same,
(1)  opposite angles are congruent
(2) opposite sides are congruent
(3) the diagonals bisect each other

Thus, the required point that is not always true for parallelograms is the diagonals are perpendicular.

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Satisfying all the given situations, there were 156 candidates in the examination.

We can use the principle of inclusion-exclusion to solve the problem.

The principle of inclusion-exclusion states that the total number of elements in the union of two or more sets is equal to the sum of the number of elements in each set, minus the number of elements in their intersection.

Here, let A be the set of candidates who offered biology, B be the set of candidates who offered history, and C be the set of candidates who offered mathematics.

Using the principle of inclusion-exclusion, the total number of candidates, N, can be found as follows:

N = (A U B U C) = (A + B + C) - (A ∩ B + B ∩ C + A ∩ C) + (A ∩ B ∩ C)

where A U B U C is the union of the three sets, A ∩ B is the intersection of A and B, and so on.

Given that:

|A| = 52, |B| = 60, |C| = 96

|A ∩ B| = 21, |A ∩ C| = 22, |B ∩ C| = 16

|A ∩ B ∩ C| = 7

Therefore,

N = (52 + 60 + 96) - (21 + 22 + 16) + 7

N = 156

Hence, there were 156 candidates in the examination.

The problem seems incomplete, it must have been

"In a certain examination, 52 candidates offers biology,60 offers history,96 offers mathematics, if 21 offered both biology and history,22 offered mathematics and biology, and 16 offered mathematics and history. If 7 candidates offered all the subject. how many candidates were there for the examination​?"

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The expression that represents the shaded length of the number line would be = 3/20 × 9/20. That is option C.

A number line is defined as the graduates line that is used to represent the position of both positive and negative real numbers.

The shaded length of the number line contains 20 bars with a total of 9 shaded bars.

There are a total number of 3 per each bar of the number line, therefore, the shaded length of the number line would be = 3/20 × 9/20.

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Answer:

r.a) (3) = r(a(3)) = r(3) = -2(3)² - 2 = -2(9) - 2 = -18 - 2 = -20

(q-r) (3) = q(3) - r(3) = -3 + 1 - (-18) = -2

So, (r.a) (3) = -20 and (q-r) (3) = -2

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

Answer: 3

Step-by-step explanation:

5x = 15

Divide both sides by 5:

x = 3 :>

[tex]\huge\text{Hey there!}[/tex]

[tex]\mathsf{5x = 15}[/tex]

[tex]\textbf{DIVIDE 5 to BOTH SIDES}[/tex]

[tex]\mathsf{\dfrac{5x}{5} = \dfrac{15}{5}}[/tex]

[tex]\textbf{SIMPLIFY it}[/tex]

[tex]\mathsf{x = \dfrac{15}{5}}[/tex]

[tex]\mathsf{x = 3}[/tex]

[tex]\huge\text{Therefore, your answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{x = 3}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

The difference between the polynomials to subtract the terms with the same degree in order to calculate the difference between two polynomials.

For instance, you would deduct 4x2 from 3x2 (which is -x2) and then deduct 2x from 5 to find the difference between 4x2 + 2x - 7 and 3x2 + 5. (which is 3x). -x2 + 3x - 7 is the difference between the two polynomials.

A polynomial is a type of equation where each variable's exponent must be a whole number.

Expressions of the type polynomials exist. Sums (and differences) of polynomial "terms" are polynomials.

To be a polynomial term, any variables in the expression must have whole-number powers (or else the "understood" power of 1, as in x1, which is normally written as x)

A polynomial term can also be a simple number. In particular, an expression must not contain any square roots of variables, any fractional or negative powers on the variables, and any variables in any fractions' denominators in order to qualify as a polynomial term.

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Therefore, there are no solutions to the given quadratic equation. 57 and −2. Yes, there are two real roots in the given equation.

The formula for determining the roots is x = (-b  (b2 - 4ac))/2a. D = b2 minus 4ac is the discriminant. The equation has two real and distinct roots if D is greater than zero.

As a result, this equation does not have any actual roots.

Quadratic equations can be solved in three basic ways: completing the square, using the quadratic formula, and factoring.

When a given quadratic equation is multiplied by ax 2 + bx + c=0, we obtain a=2, b=7, and c=5.

Now, x x x using the quadratic formula.

= \s2×3 \s−7± \s7 \s2 \s −4×3×−5

= \s6 \s−7± \s49+60

= \s6 \s−7± \s109

= \s6 \s−7+ \s109

​ \s and x= \s6 \s−7− \s109

=0.57 and x=−2.91

Therefore, 0.57 and 2.91 are the answers to the given quadratic equation.

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Full Question = Solve the following equations.

3x 2 +7x−5=0

Show all your working and give your answer correct to 2 decimal places.

If two of the angles and one of the sides of an oblique triangle are known, the Law of Sines can be used to find the missing lengths or angle measurements.

We cannot utilize the formulas specified for right triangles to solve oblique triangles; instead, new formulas must be used. We'll look at how the Law of Sines can be applied to resolve oblique triangles.

The Sine Rule:

The ratios of the length of a side to the sine of the angle opposite the side must all be the same if A, B, and C are the measurements of the angles of an oblique triangle and a, b, and c are the lengths of the corresponding sides.

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]

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The domain of function, (cd)(x) is all real values of x except x = 2, i.e., (−∞,2)∪(2,∞).

The domain of a function is defined as a set of values accepted by function. In this case, we have a rational function. A rational function will only be defined if the denominator is not equal to zero. We have, two functions, c(x) = 5/ x - 2 and

d(x) = x + 3. We should determine the domain of (cd) (x). The product of two is (cd) (x) = c(x) . d(x)

=> (cd)(x) = (5/(x - 2) )(x + 3) = 5(x+3)/(x -2)

Which is a rational function. So, the function will be defined if the denominator is not equal to zero, i.e., x - 2 ≠ 0 => x ≠ 2. Thus, domain set of function contains all real numbers values except 2. Therefore, the required domain of function (cd) (x) is all real values of x except 2.

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Complete question:

If c(x) = 5 / x - 2 and d(x) = x + 3. What is the domain of (cd)(x)?

The slope of the equation y=-2x+10 is -2.

The slope of a line is a measure of its steepness and direction. In the equation y = -2x + 10, the slope is -2.

To understand this, let's first look at the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

In this case, the equation is already in the slope-intercept form and we can see that the slope is -2. The negative sign indicates that the line is sloping downward. The bigger the absolute value of the slope, steeper the line is.

In other words, the slope of a line tells us how much y changes for each unit of change in x. In this case, the slope is -2, which means that for each unit increase in x, y decreases by 2 units.

It's also worth noting that, if the slope is positive, the line will rise from left to right, if the slope is negative, the line will fall from left to right.

Therefore, The slope of the equation y=-2x+10 is -2.

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Answer: When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function f(x)=5−3x2 f ( x ) = 5 − 3 x 2 can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

Step-by-step explanation:

Functions is an important branch of math, which connects the variable x with the variable y. Functions are generally represented as y = f(x) and it states the dependence of y on x, or we say that y is a function of x.

Yes, the anti-logarithm which is also called an antilog is the inverse of the logarithmic transform.

The antilogarithm, also referred to as an antilog, is the inverse of the logarithm transform. Since the base-10 logarithm of 1000 is 3, the anti-logarithm of 3 is 1000. To find the anti-logarithm of a base-10 logarithm, raise it by ten.

We are aware that an exponential is the inverse of a log function. We know that f(x) = log sub(x) has the inverse, f(y) = b, as a result (y). When working with the natural log, the inverse of f(x) = ln(x) is f-1(y) = ey if the base is e.

A logarithm's and an antilog's basis is 2.7183. The natural logarithm and antilog should be calculated by multiplying the logarithm and antilog, whose bases are 10 and 2,303, respectively.

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The measure of an angle is  5π/12 radians

Let us assume that the complement of the angle measures x degrees.

We have been given the measure of the angle is five times its complement.

So we get the measure of the angle = 5 × x

We know that two angles are complementary when their measures add to 90 degrees (π/2 radians).

So from the definition of complementary angles we get an equation,

x + 5x = π/2

We solve above equation to find the value of x.

x + 5x = π/2

6x = π/2

x = π/12 radians

This means, the measure of complement of angle is π/12 radians.

So, the measure required angle would be,

5 × π/12 = 5π/12 radians

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The comple question is:

An angle is equal to 5 times its complement determine its measure?.

The 12th term of the geometric  sequence is -341796875.

The common ratio (r) is:

[tex]\frac{a_{1} }{a_{2} } =-\frac{35}{7} =-5[/tex]

[tex]\frac{a_{3} }{a_{2} } =\frac{175}{-35} =-5[/tex]

(r)=-5

The sum is :

[tex]S_{n}[/tex]=a·{[tex]\frac{1-r^{n} }{1-r^{} }[/tex]}

[tex]S_{3}[/tex]= 147

So. The general form is:

[tex]a_{n}[/tex]=7·-[tex]5^{n-1}[/tex]

Now, The nth term is:

[tex]a_{1}[/tex]=7

[tex]a_{2}[/tex] = [tex]a_{1}[/tex]·[tex]r^{n-1}[/tex] = 7· [tex]-5^{2-1}[/tex] = 7 · [tex]-5^{1}[/tex] = 7 · -5 = -35

[tex]a_{3}[/tex] = [tex]a_{1}[/tex]·[tex]r^{n-1}[/tex] = 7· [tex]-5^{3-1}[/tex] = 7 · [tex]-5^{2}[/tex] = 7 · 25 = 175

[tex]a_{4}[/tex] = [tex]a_{1}[/tex]·[tex]r^{n-1}[/tex] = 7· [tex]-5^{4-1}[/tex] = 7 · [tex]-5^{3}[/tex] = 7 · -125 = -875

[tex]a_{5}[/tex] = 4375

[tex]a_{6}[/tex] = -21875

.......  so, [tex]a_{12}[/tex] = - 341796875

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If the product is positive integers is 32118, then the king's age is 53, his number of children is 6 and is length of boat is 101m.

What is integer?

The Latin term "Integer," which implies entire or intact, is where the word "integer" first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as integers.

The puzzle demands for three numbers - x, y, z which represent the king's number of children, age and length of boat respectively.

It will be advantageous to conceive the problem thus: We have but one unknown; this unknown, however, is not a number but a tripartite unknown, a triplet (x, y, z) of numbers.

It is very important to split the condition that is expressed by the statement of the problem into appropriate clauses. This needs careful con-sideration of details and considerable regrouping. After several trials (which we skip to save space) we may arrive at the following two clauses:

(r1), x, y, and z are positive integers different from 1 and such that -

xyz = 32118

(r2) 4 ≤ x < y < 100

Begin with (r1) which leaves only a finite number of possibilities, whereas (r2), which does not restrict z at all, leaves an infinite number.

Therefore, we examine (r1). Now, 32118 is divisible by 6, and so we easily decompose it into prime factors:

32118 = 2 × 3 × 53 × 101

For a decomposition into three factors we have to combine two of the four primes. Therefore, there are only six different ways to decompose the number 32118 into a product of three factors all different from 1:

6 × 53 × 101

3 × 101 × 106

3 × 53 × 202

2 × 101 × 159

2 × 53 × 303

2 × 3 × 5353

Of these six possibilities, the remaining requirement (r2) rejects all except the first one, and so we obtain -

x = 6 , y = 53 and z = 101.

Therefore, the values of x, y and z are 6, 53 and 101 respectively.

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The descriptors that are applicable for factor by grouping is 4 terms and 3 terms.

What is group factorization?

Factorization by grouping requires that we first group the terms according to their common factors. Method of factorization by term grouping:

(i) A factor can be extracted from each group of the supplied expression's groupings.

(ii) Factorize each group.

(iii) Next, eliminate the factor shared by the newly created group.

To factorize 4 terms use the method -

The first step is to group the first and last two terms together.

Then create a GCF by factoring each individual binomial.

Factor out the common binomial in step three.

To factorize 3 terms use the method -

Factoring out the GCF of the full phrase is the first step (from all 3 terms). There might not always be a GCF to account for (that is, the GCF is 1).

Next, select two terms to think about jointly (we may need to split a term into two parts). Subtract the two terms' GCFs.

Consider the remaining terms along with the factored pair and determine if there is any additional factoring that can be done (if anything).

A binomial cannot be factored by factor by grouping method as it contain only two terms which doesn't require further factoring using grouping method.

Both terms are not always perfect squares, there can be times when the factors are written as roots.

Therefore, descriptors are 4 terms and 3 terms.

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In the last basketball game, the ratio of baskets made to the total shots taken was 4:7 and if the team shot a total of 56 baskets in the game then the total basket they made is 40.

The ratio is 4:7, this means that for every 7 shots taken, 4 of them were made. If the team shot a total of 56 shots in the last game, then they made

(4/7) ×56 = 40 baskets.

This calculation can be broken down into simpler terms. For every 7 shots, 4 shots were made. Therefore, for 56 shots, 40 shots were made. Therefore, the team made 40 baskets in the last game.

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