ln(x+2)-ln(4x+3)=ln(1/2*x)

Answers

Answer 1

ln(x+2)-ln(4x+3)=ln(1/2*x)

Using properties of logarithms

[tex] \frac{ln(x+2)}{ln(4x+3)} = ln \frac{x}{2} \\ \\ \frac{x+2}{4x+3} = \frac{x}{2} \\ \\2(x+2)=x(4x+3) 2x+4=4x^2+3x \\ \\ 4 x^{2} +x-4=0 \\ \\ x= \frac{-b+/- \sqrt{b^{2}-4ac} }{2a} \\ \\ x= \frac{ -1+/-\sqrt{1+64} }{8} \\ \\ x_{1} = \frac{-1+ \sqrt{65} }{8} \\ \\ x_{2} = \frac{-1- \sqrt{65} }{8} Check: When you substitute x_{2} into \\ \\ ln(4x+3)=ln(4* \frac{-1- \sqrt{65} }{8} ) =ln( \frac{-1- \sqrt{65} }{2} ) you will get negative number under ln, that is impossible , [/tex]

so x2 is not a solution of this logarithmic equation.

Only x1 is a solution.

Related Questions

Miss Nelson Has a rectangular flower box that is 5 ft long 2 ft tall she wants the width of the box to be no more than 5 ft if the width is a whole number what are the possible volumes for the flower box

Answers

When u multiply 5•2•5 it will be 50ft of the flower box

The soup can is 6 cm tall and has a radius of 3.5cm. If you were to pull the label off the can in one complete piece,what would the area of the label be ? Use 22/7 for p.
Please help !!!

A)22 sq.cm
B)42 sq.cm
C)132 sq.cm
D)152 sq.cm

Answers

Answer is C 132cm

3,14*7=21.98
21.98*6=131,88≈132

The soup can is 6 cm tall and has a radius of 3.5cm. If you were to pull the label off the can in one complete piece,what would the area of the label be ? Use 22/7 for p.

Please help !!!

A)22 sq.cm

B)42 sq.cm

C)132 sq.cm

D)152 sq.cm

Solution:

Radius of cylindrical soup can= 3.5 cm

Height of cylindrical soup can= 6 cm

Area of cylinder =2πrh

So, Area of label of cylindrical soup can=2πrh

Plugging in the value of r and h in the formula

Area of label=2*π*3.5*6

Multiplying the constants, we get

Area=2*π*21

Area=42π

Plugging in the value of π

Area=42*[tex]\frac{22}{7}[/tex]

Multiplying the numerators

Area=[tex]\frac{924}{7}[/tex]

Area=132 sq.cm

Answer: Option (C)

Area of label= 132 sq. cm

(-8,7),(-9,-5) write in Ax+By=C

Answers

[tex]\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ &&(~ -8 &,& 7~) &&(~ -9 &,& -5~) \end{array} \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-5-7}{-9-(-8)}\implies \cfrac{-5-7}{-9+8}\implies 12 \\\\\\ \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-7=12[x-(-8)]\implies y-7=12(x+8) \\\\\\ y-7=12x+96\implies -12x+y=103[/tex]

hello can you please help me posted picture of question

Answers

Answer:
[tex] \frac{25-y^2}{16} + \frac{y^2}{9} = 1[/tex]

Explanation:
For the first given equation, we will need to isolate the x². This can be done as follows:
x² + y² = 25
x² + y² - y² = 25 - y²
x² = 25 - y² ..................> I

For the second given equation:
We will remove every x² and substitute with its equivalence from I as follows:
[tex] \frac{x^2}{16} + \frac{y^2}{9} = 1[/tex]

[tex] \frac{25-y^2}{16} + \frac{y^2}{9} = 1[/tex]

Hope this helps :)

It is the c)25-y^2-y^2 by 16-9 = 1

In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. find the probability that the number drawn is a multiple of 77 or a multiple of 4.

Answers

Total numbers = 1-25 = 25 numbers

Multiples of 77: 1, 7, 11
Multiples of 4: 1, 2, 4

Therefore, multiples of 77 or 4 are: 1, 2, 4, 7, 11 = 5 numbers

P (multiples of 7 or 4) = 5/25 = 1/5 = 0.2

please help me :)))))

Answers

Part a)

Answer: 5*sqrt(2pi)/pi

-----------------------

Work Shown:

r = sqrt(A/pi)
r = sqrt(50/pi)
r = sqrt(50)/sqrt(pi)
r = (sqrt(50)*sqrt(pi))/(sqrt(pi)*sqrt(pi))
r = sqrt(50pi)/pi
r = sqrt(25*2pi)/pi
r = sqrt(25)*sqrt(2pi)/pi
r = 5*sqrt(2pi)/pi

Note: the denominator is technically not able to be rationalized because of the pi there. There is no value we can multiply pi by so that we end up with a rational value. We could try 1/pi, but that will eventually lead back to having pi in the denominator. I think your teacher may have made a typo when s/he wrote "rationalize all denominators"

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

Part b)

Answer: 3*sqrt(3pi)/pi

-----------------------

Work Shown:

r = sqrt(A/pi)
r = sqrt(27/pi)
r = sqrt(27)/sqrt(pi)
r = (sqrt(27)*sqrt(pi))/(sqrt(pi)*sqrt(pi))
r = sqrt(27pi)/pi
r = sqrt(9*3pi)/pi
r = sqrt(9)*sqrt(3pi)/pi
r = 3*sqrt(3pi)/pi

Note: the same issue comes up as before in part a)

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

Part c)

Answer: sqrt(19pi)/pi

-----------------------

Work Shown:

r = sqrt(A/pi)
r = sqrt(19/pi)
r = sqrt(19)/sqrt(pi)
r = (sqrt(19)*sqrt(pi))/(sqrt(pi)*sqrt(pi))
r = sqrt(19pi)/pi

Comment
The radius can be found by taking the square root of A divided by pi under the root sign.

Formula
r = sqrt(A / pi)
r = sqrt(A * pi/( pi * p)i)
the root of pi^2 is pi
r = sqrt(A * pi)/pi

Problem A
r = √(50 * π) / π <<<< answer

Comment
For each problem, all you need do is put in the given areas in A B C. I don't know what your input will do with the square root sign; you can write sqrt(50 * pi) / pi

B
r = √(27*π) / π

C
r = √(19* π) / π

hello can you please help me posted picture of question

Answers

The discriminant help us to identify the nature of the roots.

If Disc > 0, the function has two distinct roots
If Disc = 0, the function has repeated roots
If Disc < 0, the function has complex roots

The turning away of the function indicates a repeated root at that point.
Thus, the given quadratic equation has a repeated root at x = 2. So the discriminant of the function is zero.

Therefore, the correct answer is option B

The problem is in the picture

Answers

Answer:
∠1 and ∠2,
∠3 and ∠4,
∠5 and ∠6

Explanation:
Supplementary angles are angles opposite each other that share the same line that is being dissected by a perpendicular line. The sum of their angles will always equal 180 degrees.

Angles 1 and 2 are an example of such supplementary angles. They share the horizontal line but are being dissected by descending line which creates their angles.

Angles 3 and 4 are more of the same. They share the somewhat-vertical line and are dissected by a different descending line that created them.

Lastly, Angles 5 and 6 are also supplementary angles. They share the somewhat-vertical line and are dissected by the horizontal line, resulting in their angles.

Angles 7 and 8 are not supplementary angles. Rather, they are vertically-opposite angles, or vertical angle pairs, two angles lying opposite each other existing at the point where two lines intersect. These angles have the same exact measurement and their sum will never equal 180 degrees.

Question 11 please solve

Answers

You will need 4 segments to connect adjacent points, and another additional two segments to connect opposite points.

We can conclude that you will need 6 segments to connect each point to every other point.

8 segments
4+2= 8
the answer is 8 4 plus 2 equals 8 even if the circle has four dots you draw four more which equals 8 either one

How to find the of each figure #5,6,7,8

Answers

5. The base area = (1/2)(6.1 m apothem)(6*7 m perimeter) = 128.1 m^2. The height = 8 m, so the volume of a pyramid = (1/3)bh = 341.6 m^3.
6. The base area = (8 yd)(10 yd) = 80 yd^2. The height = 11 yd, so the volume = (1/3)bh = 293.3 yd^3.
7. The lateral area is (11)(10) for the slanted surface, (11)(6) for the vertical rectangular surface, and (1/2)(8)(6) for each of the triangular surfaces. This is a lateral area of 110 + 66 + 2(24) = 224 in^2.The bottom surface is (11)(8) = 88, so the total surface area is 312 in^2.
8. The lateral area is (5 triangles)(1/2)(7.3 yd height)(6 yd base) = 109.5 sq. yd.The base is (1/2)(4.1 yd apothem)(5*6 yd perimeter) = 61.5 sq. yd. This is a total surface area of 109.5+61.5 = 171 sq yd.

Hello,
Please, see the attached files.
Thanks.

Mr zucco baked a total of 270 cupcakes and muffins. After ue sold 1/3 of the cupcakes and baked 18 more miffins, he had twice as many cupcakes as muffins. How many cupcakes and muffins did mr. Zucco bake to begin with?

Answers

Let cupcakes = c and muffins = m.
c + m = 270
After 1/3 of cupcakes are sold (leaving 2c/3) and 18 more muffins are baked (m + 18), there are twice as many cupcakes as muffins:
2c/3 = 2(m + 18)
2c/3 = 2m + 36
c = 3m + 54
Substituting into the original equation:
3m + 54 + m = 270
4m = 216
m = 54 muffins
c = 3(54) + 54 = 216 cupcakes

After the hairdresser Jenny had 27 centimeters cut off her hair how many decimeters of hair did jenny have cut off

Answers

Answer:

2.7 dm

Step-by-step explanation:

You want to know the number of decimeters in 27 centimeters.

SI Prefixes

The SI prefix "deci-" means 1/10.

The SI prefix "centi-" means 1/100.

This tells you that 10 cm = 1 dm.

27 cm = (27 cm) × (1 dm)/(10 cm) = 2.7 dm

Jenny had 2.7 dm of hair cut off.

A restaurant uses 290 pounds of onions every 2 and a half weeks. If they continue to use onions at the same rate, how many pounds of onions will they use in 6 weeks?

Answers

290/2.5 = 116 pounds per week

116 * 6 = 696 pounds

To find the rate of one week, we divide the amount of onions used with every week:
290/2.5 = 116
Now that we have a single rate, we find the amount for 6 weeks:
116*6 = 696
696 pounds are used in 6 weeks

the lengths of three line segments are 4 centimeters,5 centimeters,and 9 centimeters?is it 1,

Answers

9 is the base of the triangle 4 is the leg and 5 is the

No triangles can be constructed from line segments of 4 cm, 5 cm, and 9 cm in length, because the Triangle Inequality Theorem requires the sum of any two sides to be greater than the third side, which is not the case here.

The lengths of three line segments are 4 centimeters, 5 centimeters, and 9 centimeters. To determine how many triangles can be constructed with these segments as sides, we need to apply the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, the sum of the two shortest sides, 4 cm and 5 cm, is 9 cm, which is equal to the length of the longest side. Because one condition for the Triangle Inequality Theorem is not strictly being met (the sum must be greater, not equal), we can conclude that no triangles can be constructed using these three segments as sides.

When constructing triangles from three measures of angles or sides, it's important to realize that these conditions determine whether you can create a unique triangle, more than one triangle, or no triangle at all.

Linda needs 20 gallons of juice that has 75%fruit juice. she has a mixture that has 80% fruit juice and another mixtures that has 60% fruit juice. how much of each mixture should she use to get what she need?

Answers

Linda should use ...
15 gallons of 80% juice
5 gallons of 60% juice

_____
Let g represent the number of gallons of 80% juice needed. Then (20-g) is the number of gallons of 60% juice Linda will use. The amount of juice in the mixture can be written as
0.80g +0.60(20 -g) = 0.75×20
0.20g = 20(0.75 -0.60) = 20×0.15
g = 20×0.15/0.20 = 15
The amount of 80% juice needed is 15 gallons, so 5 gallons of 60% juice are needed.

What is the surface area of a rectangular prism that has a length of 9 feet, a width of 5 feet, and a height of 7.5 feet?

Answers

A suitable formula for the surface area of a rectangular prism is ...
A = 2(LW +H*(L +W))
Filling in the given information, we have
A = 2((9 ft)*(5 ft) +(7.5 ft)*(9 ft +5 ft))
A = 2(45 ft² +(7.5 ft)*(14 ft)) = 2(150 ft²)

The surface area of the rectangular prism is 300 ft².

Answer: 300 ft²

Step-by-step explanation:

Lucia is not yet 80 years old. each of her sons has as many sons as brothers. the combined number of lucias sons and grandsons equals her age, and her oldest grandson is 29. how old is lucia? place your numerical answer in the corresponding answer blank.

Answers

Not enough Information. I could say 50 and it would work. I could say 68 and it would work. Any even number thats under 80 but higher than what you think her eldest son is.

17.5% as a fraction in simplest form?

Answers

[tex]17.5\%[/tex] is the same thing as [tex]0.175[/tex]. We can rewrite this as [tex]\frac{175}{1000}[/tex]. Or course this is not simplified, so we can simplify to get [tex]\boxed{\frac{7}{40}} [/tex].

Heong cut a slice of birthday cake. The slice formed the angle shown. What is the measure of the angle shown?

Answers

Do you have a picture?

A bakery has total receipts for the month of $24,150 including sales taxes. if the sales tax rate is 5%, what are the bakery's sales for the month?
a.$24,000
b.$23,000
c.$23,100
d.$22,943

Answers

the answer would be B) $23,000

Option B= $23000 is the answer.

Explanation:

Let the bakery's sales for the month be represented by = x

As given, the bakery has total receipts for the month = $24150

Rate of sales tax = 5%

This implies that total monthly sales(x) + sales tax on 'x' = 24150

[tex]x+\frac{5x}{100}=24150[/tex]

[tex]x+0.05x=24150[/tex]

[tex]1.05x=24150[/tex]

x=23000

Hence, the bakery's sale for the month is $23,000

Find the length of AB , given that DB is a median of the triangle and AC = 24.

Answers

AB=12 because medians bisect the opposing side of a vertex.

Answer:

AB = 12 units

Step-by-step explanation:

We are given the following information in the question:

DB is the median of the triangle.

AC = 24 units

Property of median of a triangle:

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.Thus, a median divides the side of triangle in two equal parts.

Thus, DB divides AC in two equal parts.

Thus, we could say:

AB = BC

We have to find the length of AB.

[tex]\text{AB} = \displaystyle\frac{AC}{2} = \frac{24}{2} = 12\text{ units}[/tex]

Thus, AB is 12 units.

Cable hangs between two poles of equal height and 37 feet apart. at a point on the ground directly under the cable and x feet from the point on the ground halfway between the poles the height of the cable in feet is

Answers

Weight= (38.55 ft)(11.7 lb/ft) = 451 pounds

Sarah is playing a game in which she rolls a number cube 20 times. The results are recorded in the chart below. What is the experimental probability of rolling a 1 or a 3?
Number on cube:1,2,3,4,5,6
Number of times event occurs:3,6,1,5,3,2
A.0.2
B.0.3
C.0.6
D.0.83

Answers

Sarah rolled a one 3 times and rolled a three 1 time in a total of 20 rolls. She rolled one or three 4 times out of 20. Therefore,

[tex] \frac{4}{20} = 0.2[/tex]

The answer is A.

5.
Find the present value of the annuity.

Amount Per Payment: $6,225

Payment at End of Each: Quarter

Number of Years: 6

Interest Rate: 8%

Compounded: Quarterly

Answers

To solve this we are going to use the present value of annuity formula: [tex]PV=P[ \frac{1-(1+ \frac{r}{n})^{(-kt)} }{ \frac{r}{n} }] [/tex]
where
[tex]PV[/tex] is the present value
[tex]P[/tex] is the periodic payment
[tex]r[/tex] is the interest rate in decimal form
[tex]n[/tex] is the number of times the interest is compounded per year
[tex]k[/tex] is the number of payments per year
[tex]t[/tex] is the number of years

We know for our problem that [tex]P=6225[/tex] and [tex]t=6[/tex]. To convert the interest rate to decimal form, we are going to divide it by 100%:
[tex]r= \frac{8}{100} [/tex]
[tex]r=0.08[/tex]
Since the interest is compounded quarterly, it is compounded 4 times per year, so [tex]n=4[/tex]. Similarly, since the payment is made at the end of each quarter, it is made 4 times per year; therefore, [tex]k=4[/tex].
Lets replace the values in our formula:
[tex]PV=P[ \frac{1-(1+ \frac{r}{n})^{(-kt)} }{ \frac{r}{n} }] [/tex]
[tex]PV=6225[ \frac{1-(1+ \frac{0.08}{4})^{(-(4)(6)} }{ \frac{0.08}{4} }] [/tex]
[tex]PV=117739.19[/tex]

We can conclude that the present value of the annuity is $117,739.19

You randomly select one card from a 52-card deck. find the probability of selecting a black seven or a red three.

Answers

Number of cards in a deck = 52
Number of back sevens = 2
Number of red three = 2

P(Black seven or red three) 4/52 = 1/13

Answer: 1/13

The probability of selecting a black seven or a red three will be 1/13.

What is Probability?

The term probability refers to the likelihood of an event occurring.

Given that;

You randomly select one card from a 52-card deck.

Now,

Number of black sevens = 2

Number of red three = 2

So, The probability of selecting a black seven or a red three is;

= (2 + 2) / 52

= 4 / 52

= 1 / 13

Thus, The probability of selecting a black seven or a red three will be 1/13.

Learn more about the probability visit:

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The graph of the function f(x)=4sqrt x is shifted down by three units. What is the domain of the resulting function

Answers

A vertical shift changes the range, but not the domain. The domain is still ...
x ≥ 0.

11 black balls and 14 white balls are placed in an urn. two balls are then drawn in succession. what is the probability that the second ball drawn is a white ball if the second ball is drawn without replacing the first ball?

Answers

[tex] |\Omega|=25\cdot24=600\\
|A|=11\cdot14+14\cdot13=154+182=336\\\\
P(A)=\dfrac{336}{600}=\dfrac{14}{25}=56\% [/tex]

To test upper h 0: muequals50 versus upper h 1: muless than50, a random sample of size nequals22 is obtained from a population that is known to be normally distributed. complete parts (a) through (d) below.

Answers

Final answer:

The question relates to a statistical hypothesis test, specifically comparing a sample mean to a known population mean. This involves recognizing null and alternative hypotheses, identifying and calculating a test statistic, deciding a significance level, determining a critical value, and finally comparing the test statistic to this critical value to decide whether to reject or not reject the null hypothesis.

Explanation:

The subject matter posed in the question concerns a statistical hypothesis test.

In these cases, we have two contradictory hypotheses about a population: the null hypothesis, denoted as 'H₀', which generally represents the status quo or a statement of no effect (e.g. a population mean is equal to 50), and the alternative hypothesis, denoted 'H₁', the contrary of the null hypothesis (population mean is not equal to 50). In your specific case, the null hypothesis 'H₀' is that the mean is equal to 50 (μ = 50) and the alternative hypothesis 'H₁' is that the mean is less than 50 (μ < 50).

To test these hypotheses, you could follow these steps:

Identify a test statistic: Since we know the population is normally distributed and we are comparing the sample mean to the population mean, we can use a z-test statistic.

Compare the test statistic to the critical value to decide whether to reject or fail to reject the null hypothesis.

After these steps, interpret the results in terms of the problem.

Learn more about Hypothesis Testing here:

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What is the perimeter of the rectangle? Write your answer in scientific notation. The area is 2.76 x 10 pw12
the width is 4.6 x 10 pw5
pw is the power

Answers

Final answer:

To calculate the perimeter of a rectangle given the area and the width, divide the area by the width to find the length, then use the formula Perimeter = 2(Length + Width), adjusting for scientific notation as necessary. This will provide the perimeter in scientific notation.

Explanation:

To find the perimeter of the rectangle when we have the area and the width, we need to first calculate the length of the rectangle using the formula Area = Length × Width. The width is given as 4.6 x 105 and the area is 2.76 x 1012. To find the length, we divide the area by the width:

Length = Area ÷ Width = (2.76 x 1012) ÷ (4.6 x 105)

After performing the division, we obtain the length in scientific notation. The next step is to use the formula Perimeter = 2(Length + Width). After calculating the sum of the length and width, we multiply by 2 to obtain the perimeter of the rectangle in scientific notation.

Let's assume after performing the calculation that the length is found to be L x 10n where L and n are specific numbers. The perimeter now would be calculated as:

Perimeter = 2((L x 10n) + (4.6 x 105))

Be sure to adjust the exponents so they match before adding the length and width together. This might involve converting one or the other so that you can add them directly. Once that is done, you can multiply by 2 to find the total perimeter of the rectangle and express it in scientific notation.

The hyperbola (x-5)^2/7 - (y+3)^2/9 = 1 is shifted to the right by 4 units and upward by 3 units. the new center of the hyperbola is

Answers

The center of the given hyperbola is (5, -3). Since it is shifted by (4, 3), the new center will be
(5, -3) +(4, 3) = (9, 0)

Answer:

( 9 ,0)

Step-by-step explanation:

Given : [tex]\frac{(x -5)^{2}}{7} - \frac{(y+3)^{2}}{9} = 1.[/tex] is shifted to the right by 4 units and upward by 3 units

To find : New center of the hyperbola .

Solution : We have given

[tex]\frac{(x -5)^{2}}{7} - \frac{(y+3)^{2}}{9} = 1.[/tex]

Center of hyperbola is ( 5 , -3)

By the transformation rule f(x) →→ f(x -h) + k it mean f(x) is shifted to right by h unit and k unit up.

Then Center of hyperbola is shifted to the right by 4 units and upward by 3 units.

( 5 , -3) →→ (5 + 4 , -3 + 3

( 5 , -3) →→ ( 9 ,0)

Therefore, new center is ( 9 ,0).

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A lab technician worked 48 3/4 hours over a six day period. If she worked the same number of hours each day, how many hours did she work each day? Find the value of x. suppose you drop a tennis ball from a height of 15 feet. After the ball hits the floor, it rebounds to 85% of its previous height. how high will the ball rebound after its third bounce? round to the nearest tenth Pollution that falls out of the atmosphere that originated from a factory is nonpoint, or secondary source, pollution with the factory being the point, or primary source, of the pollution. Is acid rain a point source (primary) or nonpoint source (secondary) of pollution? What is the effect of acid rain on the environment? (Site 1) two people are pushing a couch from opposite sides. one person pushes with a force of 2 N. the other person pushes with a force of 4 N. which of the following best describes how the couch will move? Which two principles have directed the foreign policy of the United States since world war 2 what is the area of a circle with a circumference of 12pi? Which state(s) of matter have no definite shape or volume? Which of the following federal agencies prohibited advertisers from broadcasting commercials at a higher volume than the televised program during which they appear? cheap is to valuable as common is to? The formula for the area of a trapezoid is A = h(b1 + b2), where A = area, h = height, b1 = base 1, and b2 = base 2. For a trapezoid with a height of 7 centimeters and base lengths of 6 centimeters and 10 centimeters, the area is square centimeters Four-year-old Alice has a known peanut allergy. What is a proper step her childcare center can take to avoid a life-threatening emergency?A)separate Alice so she eats alone at snack timeB)tell Alice she cannot eat any peanutsC)explain to Alices parents that she may not attend childcare due to her allergyD)read all labels and only allow snacks without peanuts within the childcare environment the following data are the numbers of customers at a coffee shop over a 10 day period 135,124,140,122,409,132,119,128,136,125 which measure of center best represents the daily average of customers at the coffee shop? What is the very large depression at the top of some volcanoes called? An item is called "price elastic" if you drop the price and revenue goes up. Last week you sold 10 sodas for $2 each; this week you sold 30 sodas for $1 each. Is soda price elastic? The parts of the pancreas that have endocrine functions are known as the _____.a. parathyroid glandsb. adrenal glandsc. pancreatic isletsd. endocrine glands Which city became the center of the catholic church in the early 1300s? what is the height of a building in meters if it takes a rock 3.8 seconds to drop from its roof Who advocated for the immediate and uncompensated release of all slaves held in the united states? Why did the Renaissance begin in Italy? Choose all answers that are correct. A. Italy's central location provided a good setting for commerce, so trade revived there first. B. The Black Plague had not devastated Italy as it had other areas of Europe. C. Italy had been the center of the Roman Empire and the ancient ruins reminded people of Rome's greatness. D. The Middle Ages had little impact on Italy where centralized government and the economy continued to flourish after the fall of Rome.