What are the real-number solutions of the equation 2x5 = 12x3 - 16x ?

Answer:

inverse property

To find h'(x), differentiate the function h(x) = (cos x)/f(x) using the product rule.

To find h'(x), we need to differentiate the function h(x) = (cos x)/f(x).

First, let's find the derivative of cos x, which is -sin x.

Next, we need to find the derivative of f(x). Since f(π/3) = 3 and f '(π/3) = −7, we know the slope of the tangent line at x = π/3 is -7.

Using the product rule, we can now differentiate h(x) = (cos x)/f(x) as follows:

h'(x) = [f(x)(-sin x) - cos x(f '(x))]/[f(x)]^2

By squaring both sides and solving for [tex]\(x\),[/tex] we find [tex]\(x = 4\)[/tex] as the solution, which is not extraneous upon substitution into the original equation.

To solve the equation [tex]\(\sqrt{2x + 1} = 3\),[/tex] we need to isolate[tex]\(x\).[/tex] Here's how:

1. Square both sides of the equation to eliminate the square root:

[tex]\[ (\sqrt{2x + 1})^2 = 3^2 \][/tex]

[tex]\[ 2x + 1 = 9 \][/tex]

2. Subtract 1 from both sides to isolate [tex]\(2x\)[/tex]:

[tex]\[ 2x = 9 - 1 \][/tex]

[tex]\[ 2x = 8 \][/tex]

3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:

[tex]\[ x = \frac{8}{2} \][/tex]

[tex]\[ x = 4 \][/tex]

Now, we have [tex]\(x = 4\)[/tex]. To determine if it's an extraneous solution, we need to check if it satisfies the original equation.

Substitute [tex]\(x = 4\)[/tex] into the original equation:

[tex]\[ \sqrt{2(4) + 1} = 3 \][/tex]

[tex]\[ \sqrt{9} = 3 \][/tex]

[tex]\[ 3 = 3 \][/tex]

Since the equation holds true, [tex]\(x = 4\)[/tex]is a valid solution, not an extraneous one.

Therefore, the solution to the equation [tex]\(\sqrt{2x + 1} = 3\) is \(x = 4\),[/tex]and it is not an extraneous solution.

Answer:  The correct option is (B) 3.

Step-by-step explanation:  Given that Lily  just paid off a $400 loan. She had to pay $60 in interest at a simple annual interest rate of 5%.

We are to find the number of years for which Lily had this loan.

Let n be the required number of years.

Also, P = $400, S.I. = $60 and r% = 5%.

Therefore, by the formula of simple interest, we have

[tex]S.I.=\dfrac{Prn}{100}\\\\\\\Rightarrow 60=\dfrac{400\times5\times n}{100}\\\\\\\Rightarrow n=\dfrac{60}{20}\\\\\\\Rightarrow n=3.[/tex]

Thus, the required number of years is 3.

Option (B) is CORRECT.

Answer:

[tex]3\frac{5}{9}[/tex] cups of milk and [tex]\frac{1}{3}[/tex] cups of oil for 8 people .

Step-by-step explanation:

Cups of milk for 6 people = [tex]2 \frac{2}{3} =\frac{8}{3}[/tex]

Cups of milk for 1 people = [tex]\frac{\frac{8}{3}}{6}=\frac{4}{9}[/tex]

Cups of milk for 8 people = [tex]\frac{4}{9} \times 8= \frac{32}{9}[/tex]

Cups of oil for 6 people = [tex]\frac{1}{4}[/tex]

Cups of oil for 1 people = [tex]\frac{\frac{1}{4}}{6}= \frac{1}{24}[/tex]

Cups of oil for 8 people = [tex]\frac{8}{24}=\frac{1}{3}[/tex]

Hence [tex]3\frac{5}{9}[/tex] cups of milk and [tex]\frac{1}{3}[/tex] cups of oil for 8 people .

The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.

In this question, we are going to solve for [tex]x[/tex] with the help of Logarithm Properties, which are described in the image attached below.

[tex]\log x + \log (3\cdot x - 13) = 1[/tex]

[tex]\log [x\cdot (3\cdot x - 13)] = 1[/tex]

[tex]\log (3\cdot x^{2}-13\cdot x) = 1[/tex]

[tex]10^{\log(3\cdot x^{2}-13\cdot x)} = 10^{1}[/tex]

[tex]3\cdot x^{2}-13\cdot x = 10[/tex]

[tex]3\cdot x^{2}-13\cdot x -10 = 0[/tex]

This is a Second Order Polynomial, which can be solved by Quadratic Formula:

[tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex]

The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.

Please see this question related to Logarithm Properties for further details:

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Answer:

The answer is b. 5 feet

Step-by-step explanation:

From the given condition, the rate of change of radius is constant.

How quickly something evolves over time is referred to as its rate of change (ROC).

Given:

Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area.

Let V be the volume of the sphere & S be the surface area.

According to the question,

dV/dt = kS

Since,

V = 4/3πr³

dV/dt = 4πr²dr/dt

S = 4πr²

Putting these values to the above expression,

4πr²dr/dt = k4πr²

dr/dt = k

Therefore, the rate of change of radius is constant.

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Answer:

A tangent is a line, ray, or line segment that intersects a circle at exactly one point (called the point of tangency) and contains no points inside the circle. A chord is a segment with both endpoints on a circle. Tangents intersect the circle at one point, while a chord intersects at two.

I've checked this answer, E counted it as correct. Hope this helped!!!

A tangent touches a circle at one point, while a chord connects any two points on the circle's circumference.

A tangent and a chord are both important concepts in geometry, but they have distinct characteristics.

A tangent is a line that intersects a circle at exactly one point, touching the circle's circumference at that point.

It never crosses the circle. Tangents are perpendicular to the radius that intersects the point of tangency.

On the other hand, a chord is a line segment connecting any two points on a circle's circumference. Unlike tangents, chords can intersect the circle at multiple points.

The diameter is a special case of a chord that passes through the center of the circle.

Hence,

Tangents touch a circle at one point, while chords connect two points on a circle's circumference.

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The correct equation to model the rectangle's area where the length is 3 inches more than the width and the area is 70 square inches is W(W + 3) = 70, which simplifies to W^2 + 3W = 70.

The student is asking for the correct equation to model a rectangle's area where the length (L) is 3 inches more than the width (W), and the area is 70 square inches. To find an equation that models the situation, we need to express L in terms of W. Since L is 3 inches more than W, we can write L as W + 3. The area (A) of a rectangle is found by multiplying the length by the width, so A = L x W.

Therefore, the equation that models this situation is W(W + 3) = 70. To see why, let's insert the expression for L into the area formula:

A = L x W = (W + 3) x W

This simplifies to:

A = W^2 + 3W

Since we know the area A is 70 square inches, we substitute and get the equation:

W^2 + 3W = 70

Which is the correct model for the given situation.

Answer:

Angle A= Angle A' = 125°

Step-by-step explanation:

We have given that : Eric reflected parallelogram ABCD across the x-axis.

                                  If angle A is 125° and angle B is 55°

To find :  Degree measurement of angle A'

Solution :

As it is reflected parallelogram , and by property of reflection it form congruent parallelogram

since it is congruent then measures of angle remain same

by this statement the measure of angle of parallelogram ABCD

remain same or equal to parallelogram A'B'C'D'

⇒Angle A= angle A' = 125°

Answer:

Prime

Step-by-step explanation:

A prime number is a number that has only one factor. A composite number is a number that has more than one factor.

-kiniwih426

$2073.60 per month

===================

To calculate Tyrone's total monthly cash inflow, we need to determine his weekly and then monthly earnings.

Calculate weekly gross income:

Calculate net pay per week:

Calculate monthly cash inflow:

Tyrone's total monthly cash inflow is $2073.60.

The Boyce's house bought for $96,000 in 1995, increased approximately 4% yearly. By applying the compound interest formula, the house's value in 2007 would be approximately $139,254.09.

We need to use the compound interest formula to calculate the value of Boyce's house in 2007 because the house price increase is compounded annually. The formula is P(1 + r/n)^(nt). Here, P is the initial amount (i.e., the original house price), r is the annual interest rate (the rate of increase in house value), and n is the number of times interest is compounded yearly. T is the number of years the money is invested.

However, as we deal with annual compounding, the formula simplifies to P(1 + r)^t. In this case, P = $96,000, r = 4% or 0.04, and t = 2007 -1995 = 12 years.

Inserting these values, we get 96000(1 + 0.04)^12 = $139,254.09 (rounded to the nearest cent).

So, according to the 4% annual increase rate, Boyce's house would be worth approximately $139,254.09 in 2007.

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The graphs of the system of equations −5y=2x+3 and y=25x+4 are neither parallel nor perpendicular.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

It is given that:

The system of equations is:

 −5y=2x+3 and y=25x+4

The slope of parallel graphs is the same. The reciprocal slopes of a perpendicular are opposite.

These equations are neither because they have slopes of 25 and -2/5.

Thus, the graphs of the system of equations −5y=2x+3 and y=25x+4 are neither parallel nor perpendicular.

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The slope of a line through the points (3,7) and (5,7) is 0, indicating a horizontal line.

To calculate the slope of a line passing through two points, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, the points are (3,7) and (5,7). Using the slope formula, we get:

m = (7 - 7) / (5 - 3) = 0 / 2 = 0

So, the slope of the line that passes through these points is 0, which means the line is horizontal.

Answer:

Intersecting

Step-by-step explanation:

After having singled out y on one side of both equations you should have.

Y=b/a• x - 2/-a

And

Y= -a/b • x + 3/b

As you can see they have opposite reciprocals which is an intersection

Answer:

a

Step-by-step explanation:

Final answer:

Josephine will need at least 13 full weeks to start making a profit. The correct inequality that models this economic scenario is 550w > 4500 + 200w.

Explanation:

The minimum number of weeks it will take for Josephine to make a profit in her cosmetics business can be determined by setting up an inequality where the total earnings must be greater than the sum of the initial investment and the running costs. We define w as the number of weeks. Josephine earns $550 per week, so her earnings after w weeks are 550w. The initial investment is $4500 and the weekly expense is $200, so the total expenses after w weeks are 4500 + 200w. To make a profit, the earnings must be greater than the expenses:

550w > 4500 + 200w

To solve for w, we need to collect like terms:

550w - 200w > 4500

350w > 4500

Dividing both sides by 350:

w > 4500 / 350

w > 12.86

This means Josephine will need to work for at least 13 full weeks to make a profit.