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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.

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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.

$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 speed of Bicyclist A (heading north) is 16 km/h.

The speed of Bicyclist B (heading south) is 11 km/h.

Given:

- Let x be the speed of bicyclist A (heading north).

- Let y be the speed of bicyclist B (heading south).

- Bicyclist A is 5 km/h faster than Bicyclist B, so [tex]\(x = y + 5\)[/tex] (Equation 1).

- Time from 9:00 to 10:45 is 1.75 hours.

- The total distance covered by both bicyclists during this time is 47.25 km.

From the equation for time and distance, we have:

[tex]\[ x \times 1.75 + y \times 1.75 = 47.25 \][/tex]

Substituting Equation 1 into this equation:

[tex]\[ (y + 5) \times 1.75 + y \times 1.75 = 47.25 \]\[ 1.75y + 8.75 + 1.75y = 47.25 \]\[ 3.5y + 8.75 = 47.25 \]\[ 3.5y = 38.5 \]\[ y = \frac{38.5}{3.5} \]\[ y = 11 \text{ km/h} \][/tex]

Substituting the value of y back into Equation 1:

[tex]\[ x = 11 + 5 \]\[ x = 16 \text{ km/h} \][/tex]

Thus, the correct answer is:

- The speed of Bicyclist A (heading north) is 16 km/h.

- The speed of Bicyclist B (heading south) is 11 km/h.

Question :

At 9:00 on Saturday morning, two bicyclists heading in opposite directions pass each other on a bicycle path.The bicyclist heading north is riding 4 km/hour faster than the bicyclist heading south. At 10:15, they are 40 km apart. Find the two bicyclists’ rates.

Answer:

The answer is b. 5 feet

Step-by-step explanation:

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.

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:

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:

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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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

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

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

The problem can be solved by setting up the equation 0.12x = $420, where 'x' stands for your total income. Dividing both sides of the equation by 0.12 gives x = $3500 which is the total annual income. Thus, you earned $3,500 this year.

To solve this mathematical problem, you can set up an equation that represents the problem context. You know that 12% of your whole year's income equals $420. So if 'x' represents your total income, the equation becomes 0.12x = $420. To solve for 'x' (your total income), you would divide both sides of the equation by 0.12.

So, x = $420 ÷ 0.12. When you perform this calculation, you'll find that 'x' equals $3,500. Therefore, you earned $3,500 this year.

The answer to the problem is D. $3,500.

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so the answer is 800. I guarantee you this will be the answer I promise

Final answer:

Vanessa's initial deposit in the bank, earning 1.25% simple interest per year, was $800. This is calculated using the simple interest formula, rearranged to solve for the principal based on the $5 interest earned in half a year.

Explanation:

To find Vanessa's initial deposit, we use the simple interest formula, which is Interest = Principal × Rate × Time. Given Vanessa earned $5.00 in interest in half a year (0.5 years) at a rate of 1.25% per annum, we can rearrange the formula to solve for the principal (initial deposit).

Using the given information: $5 = Principal × 0.0125 × 0.5, we can solve for the Principal as follows:

$5 = Principal × 0.00625
Principal = $5 ÷ 0.00625
Principal = $800

Therefore, Vanessa's initial deposit was $800.

Answer:

inverse property

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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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.