A baseball card is originally worth 3$. It increases by 7% every year while the player is still active. After the player retires it increases 10% every year. If a player remains active for 7 years, how much is the baseball card worth after 20 years?

Final answer:

The equation 4x + 3 = 7 is solved algebraically by subtracting 3 from both sides and then dividing by 4 to find x = 1. There is no need for logarithms or the change of base formula to solve this linear equation.

Explanation:

The equation 4x + 3 = 7 is a linear equation and can be solved using simple algebraic manipulations rather than the change of base formula for logarithms. However, to show how logarithms could hypothetically be used (even though it's over-complicating this particular problem), we can take the following steps:

Divide both sides by 4: x = 1.

Therefore, the solution to the equation is x = 1. There is no need to round to the nearest thousandth as the solution is a whole number. The change of base formula, logb y = (log y) / (log b), is not necessary for this equation.

The solution to the equation [tex]\(4x + 3 = 7\) is \(x = 1\),[/tex] obtained by isolating [tex]\(x\)[/tex]

Step-by-Step Solution:

1. Write down the equation:

[tex]\[ 4x + 3 = 7 \][/tex]

2. Isolate the term with the variable [tex]\(x\):[/tex]

  - To do this, we need to remove the constant term on the left side by subtracting 3 from both sides of the equation.

[tex]\[ 4x + 3 - 3 = 7 - 3 \][/tex]

  - Simplify both sides:

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

3. Solve for [tex]\(x\):[/tex]

  - To isolate [tex]\(x\)[/tex], divide both sides of the equation by 4:

[tex]\[ \frac{4x}{4} = \frac{4}{4} \][/tex]

- Simplify both sides:

[tex]\[ x = 1 \][/tex]

Conclusion:

The solution to the equation [tex]\(4x + 3 = 7\)[/tex] is:

[tex]\[x = 1\][/tex]

ANSWER

[tex]\theta = \frac{\pi}{2} ,\frac{3\pi}{2},\frac{\pi}{4} , \frac{5\pi}{4} [/tex]

EXPLANATION

We want to solve the equation;

[tex] \cos( \theta) - \tan( \theta) \cos( \theta) = 0[/tex]

We factor to get:

[tex] \cos\theta(1 - \tan\theta) = 0[/tex]

Either

[tex]\cos\theta = 0[/tex]

Or

[tex]1 - \tan \theta = 0[/tex]

For

[tex]\cos\theta = 0[/tex]

We gave

[tex] \theta = \frac{\pi}{2} ,\frac{3\pi}{2}[/tex]

When

[tex]\tan \theta = 1[/tex]

Then,

[tex] \theta = \frac{\pi}{4} , \frac{5\pi}{4} [/tex]

Therefore the solution on the interval

[tex]0 \leqslant \theta \: \leqslant 2\pi[/tex]

is

[tex]\theta = \frac{\pi}{2} ,\frac{3\pi}{2},\frac{\pi}{4} , \frac{5\pi}{4} [/tex]

Answer:

B. 330

Step-by-step explanation:

The question indicates the order doesn't matter, so it's a combination and not a permutation.

The combinations are calculated using this formula:

[tex]C(n,r) = \frac{n!}{r! (n-r)!}[/tex]

In this case we have a population of 11 (n = 11) and a selection of 4 (r=4), so...

[tex]C(11,4) = \frac{11!}{4! (11-4)!} = 330[/tex]

So, there are 330 different combinations that can be made of 4 paintings out of a selection of 11.

Answer:

The correct answer is option B.  330

Step-by-step explanation:

It is given that,There are 11 paintings at an art show. Four of them are chosen randomly to display in the gallery window.

To find the possible ways

There are total 11 paintings.

We have to choose 4 of them

Possible number of ways = 11C₄

 = (11 * 10 * 9 )/(1 * 2* 3 * 4)

 = 330 ways

Therefore the  correct answer is option B.  330

Answer:

B) {x|-13 < x < 13}

C) {x|x < -4 or x > 4}

Step-by-step explanation:

Given in the question,

Remove the absolute value term.

If your absolute value is less than a number, then set up a three-part compound inequality that looks like this:

-(number on other side) < (quantity inside absolute value) < (number on other side)

-13 < x < 13

If your absolute value is greater than a number, then set up an "or" compound inequality that looks like this:

(quantity inside absolute value) < -(number on other side)

OR

(quantity inside absolute value) > (number on other side)

x < - 4 or x > 4

Answer:

a.<55, -27>

Step-by-step explanation:

The given vectors are u = <7, -3>, v = <-9, 5>.

We want to find 4u - 3v.

We substitute the vectors and multiply by the scalars.

4u - 3v=4<7, -3>-3 <-9, 5>.

4u - 3v=<28, -12>- <-27, 15>.

4u - 3v=<28--27, -12-15>

4u - 3v=<55, -27>

Answer:

√85

Step-by-step explanation:

See image

Distance formula can be used to find the distance between two points.

[tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}  }[/tex]

Answer: Second option

[tex]d=\sqrt{85}[/tex]

Step-by-step explanation:

The formula to find the distance between two points is:

[tex]d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]

In this problem the points are C(0, 4), T(-6, -3)

This is

[tex]x_1=0\\\\x_2=-6\\\\y_1=4\\\\y_2=-3[/tex]

So the distance is:

[tex]d=\sqrt{((-6)-0)^2 + ((-3)-4)^2}[/tex]

[tex]d=\sqrt{(-6)^2 + (-7)^2}[/tex]

[tex]d=\sqrt{36 + 49}[/tex]

[tex]d=\sqrt{85}[/tex]

Answer:

[tex]\frac{10x^2+6x}{6x^2}[/tex] and [tex]\frac{21x}{6x^2}[/tex].

Step-by-step explanation:

The given expression is:

[tex]\frac{5x+3}{3x}[/tex] and [tex]\frac{7x}{3x^2}[/tex].

The least common denominator is: [tex]6x^2[/tex].

We collect  LCM for the denominator to obtain;

[tex]\frac{2x(5x+3)}{6x^2}[/tex] and [tex]\frac{3(7x)}{6x^2}[/tex].

We multiply out the parenthesis to obtain;

[tex]\frac{10x^2+6x}{6x^2}[/tex] and [tex]\frac{21x}{6x^2}[/tex].

Therefore the correct answer is B or the second option.

Answer:

Step by step explanation

the equation is 225=27x-18

the first step is to add the 18 to the other side to get

243=27x

now divide by 27 on both sides

243/27=27x/27

solve to get

9=x

Answer:

This is the answer : (answer given in the picture)

Answer:

vertex = (- 3, 24)

Step-by-step explanation:

Given a quadratic in standard form y = ax² + bx + c : a ≠ 0, then

The x- coordinate of the vertex is

[tex]x_{vertex}[/tex] = - [tex]\frac{b}{2a}[/tex]

y = - x² - 6x + 15 ← is in standard form

with a = - 1, b = - 6, thus

[tex]x_{vertex}[/tex] = - [tex]\frac{-6}{-2}[/tex] = - 3

Substitute x = - 3 into the function for corresponding value of y

y = - (- 3)² - 6(- 3) + 15 = - 9 + 18 + 15 = 24

vertex = (- 3, 24)

Answer:

A) Δ RQP ↔ ΔUTS

Step-by-step explanation:

Answer:

Option A) is correct

Step-by-step explanation:

Given : [tex]\Delta PQR \leftrightarrow  \Delta STU[/tex]

We need to find which correspondence is equivalent to [tex]\Delta PQR \leftrightarrow  \Delta STU[/tex] .

Solution :

One to one correspondence means measure of each side and each angle of one triangle is equal to the side and angle of the other triangle .

Here, we observe that option A) is equivalent to [tex]\Delta PQR \leftrightarrow  \Delta STU[/tex] as explained below :

As per [tex]\Delta PQR \leftrightarrow  \Delta STU[/tex] ,

[tex]PQ=ST\,,\,QR=TU\,,\,PR=SU\\\angle P=\angle S\,,\,\angle Q=\angle T\,,\,\angle R=\angle U[/tex]

Also, as per [tex]\Delta RQP \leftrightarrow  \Delta UTS[/tex] , we have

[tex]PQ=ST\,,\,RQ=UT\,,\,PR=SU\\\angle P=\angle S\,,\,\angle Q=\angle T\,,\,\angle R=\angle U[/tex]

Therefore,

[tex]\Delta RQP \leftrightarrow  \Delta UTS[/tex] is equivalent to [tex]\Delta PQR \leftrightarrow  \Delta STU[/tex]

So, option A) is correct .

Answer:  D) -2 |x| - 2

Step-by-step explanation:

A v-shaped graph is an absolute value graph.  

The general form of an absolute value equation is: y = a |x - h| + k

where (h, k) represents the vertex and "a" represents the vertical stretch (aka slope).  

The vertex of the given graph is (0, 2), however the graph is inverted (upside-down) which is a reflection across the x-axis.  Therefore,

4x + 7 = 31

    -7    -7

4x = 24

/4     /4

x = 6 (Answer)

Prove.

4(6) + 7 = 31

24 + 7 = 31

31 = 31

True.

The solution to the given equation 4x + 7 = 31 is x = 6, which is determined by subtraction.

The equation is given as follows:

4x + 7 = 31

To solve the equation 4x + 7 = 31 for x, we want to isolate the variable x on one side of the equation.

First, we can begin by subtracting 7 from both sides of the equation:

4x + 7 - 7 = 31 - 7

Simplifying, we get:

4x = 24

Next, we want to isolate x, so we divide both sides of the equation by 4:

4x/4 = 24/4

This simplifies to:

x = 6

Therefore, the solution to the equation 4x + 7 = 31 is x = 6.

Learn more about the solution to the equation here:

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Okay, so there are nickels and pennies in this jar.

So, let's call the number of pennies X

Since there are 161 coins altogether, the number of nickels is (161 - X)

So now we solve the equation:

.01X + .05(161 - X) = 3.09

.01X + 8.05 - .05X = 3. 09

4.96 = .04X

124 = X

So, there are 124 pennies

Therefore, there are 161 - 124 = 37 nickels

Let's check our answer:

124 x .01 + 37x .05 = 1.24 + 1.85 = 3.09

It works!

I think it is 18

nickels for $3.09

Ari will water and weed his garden on the same day again in 21 days by finding the Least Common Multiple (LCM) of the watering and weeding cycles, which are 3 and 7 days respectively.

The question asks us to find out how many days will pass until Ari once again waters and weeds his garden on the same day. This problem can be solved by finding the Least Common Multiple (LCM) of the watering and weeding cycles. Ari waters his garden every 3 days and weeds it every Saturday, which suggests a 7-day cycle for weeding.

To find the LCM of 3 and 7, we list the multiples of each number:

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, ...

The smallest multiple they both share is 21. Therefore, Ari will water and weed his garden on the same day again in 21 days.

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Answer:

Part 1) The sum of the interior angles is equal to 180 degrees

Part 2) The measure of angle N is ∠N=60°

Part 3) The measure of angle M is ∠M=120°

Part 4) The sum of the the exterior angles is 360°

Step-by-step explanation:

Part 1) What is the sum of the interior angles of the equilateral triangle?

we know that

The sum of the interior angles of any triangle must be equal to 180 degrees

Part 2) What is the measure of ∠N?

we know that

An equilateral triangle has three equal sides and three equal internal angles ( each internal angle measure 60 degrees)

so

In this problem

∠N=60°

Part 3) What is the measure of ∠M? Explain your reasoning.

we know that

∠M+∠N=180° -----> by supplementary angles (linear pair)

we have

∠N=60°

substitute

∠M+60°=180°

∠M=180°-60°=120°

Part 4) What is the sum of the exterior angles of the equilateral triangle ∠M + ∠R + ∠X?

we know that

The sum of the the exterior angles of any polygon is equal to 360 degrees

In this problem

we have

∠M=∠R=∠X=120°

so

∠M+∠R+∠X=120°+120°+120°=360°

Answer:

Step-by-step explanation:

The sum of the interior angles is equal to 180 degrees. The measure of angle N is ∠N=60°. The measure of angle M is ∠M=120°. The sum of the the exterior angles is 360°

Answer:

Step-by-step explanation:

A]

The center is the average values of the 2 endpoints (which is also the diameter).

x:(-10 + 4)/2 = -6/2 = -3

y:(-2 + 6)/2 = 4/2 = 2

Center(-3,2)

B]

The radius is the distance from the center to one of the end points.

r^2 = (-3 - 4)^2 + (2 - 6)^2

r^2 = (-7)^2 + (-4)^2

r^2 = 49 + 16

r^2 = 65

r = sqrt(65)

C]

(x + 3)^2 + (y - 2)^2 = 65

Graph

The graph has been included so that you can see that the center I have calculated is between (-10,-2) and (4,6)

Further, it shows the the circle goes the two end points of the diameter.

Answer:

C and E

Step-by-step explanation:

Let's factor this the "old fashioned" way.  The standard form of a quadratic is

[tex]y=ax^2+bx+c[/tex]

If you're familiar with the quadratic formula I'd say throw it into that, but if not, again, let's do it the "old fashioned" way.  

We need to find the product of our a and c.  Our a = 2 and our c = -3.  So that gives us a -6.  Now we have to find the factors of 6 (the negative right now doesn't matter so much).  The factors of 6 are 1, 6  and 2, 3.  Both of those possibilities will work to give us a +5, which is the linear term.  Puttng in the 2, 3 first:

[tex]0=2x^2+3x+2x-3[/tex]

Now group the terms together into groups of 2:

[tex]0=(2x^2+3x)+(2x-3)[/tex]

The idea is to factor out something common in each term so that what's left over in the parenthesis in both terms is exactly the same.  In the first term we can factor out a common x, and in the second term, the only thing common is a 1.  So that looks like this:

[tex]x(2x+3)+1(2x-3)[/tex]

What's inside those parenthesis are not actually identical, so 2 and 3 won't work.  Lets try 1 and 6.  For those 2 numbers to equal a +5, the 6 is positive and the 1 is negative.  So let's try that:

[tex](2x^2+6x)+(-x-3)[/tex]

In the first term we can factor out the common 2x and in the second term we can factor out the common -1:

2x(x + 3) - 1(x + 3)

Now what's common is (x + 3), so we can factor THAT out and what is left over is 2x - 1:

(x + 3)(2x - 1) = 0

If x + 3 = 0, then x = -3

and if 2x - 1 = 0, then 2x = 1 and x = 1/2

Answer:

400

Step-by-step explanation:

Just take 10% of 4,000 flashlights.  400 flashlights are likely to be defective.

Answer:

1/9

Step-by-step explanation:

6/9+ 2/9= 8/9

then we subtract 8/9 from 1

1-8/9

then we have to convert 1 into a fraction

9/9-8/9

this leaves 1/9

Answer:

1/9

Step-by-step explanation:

We can make an equation that looks like this:

2/9+6/9+x/9=9/9

Let's cancel out the denominator:

2+6+x=9

Then combine like terms:

8+x=9

Then solve for x:

x=1

Then add in the denominator we canceled out:

x=1/9

Hope I helped, soz if I'm wrong!

~Potato.

Copyright Potato 2019.

P.S. Why would your friend clean your room lel :P

Answer:

11.820 ft

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you that ...

Sin = Opposite/Hypotenuse

The length of the ladder is the hypotenuse of a right triangle with 52° as the base angle. The side opposite is the height up the building where the top of the ladder rests. So, you have the relation ...

sin(52°) = height/(15 ft)

Multiplying by the denominator gives you ...

height = (15 ft)·sin(52°) ≈ 11.820 ft

___

You may need to round this number appropriately.

Answer:

A)

15 hours

B)

108 hours

C)

2074.29 miles

Step-by-step explanation:

Under the assumption the earth is a perfect circle, then in one complete rotation about its axis ( 24 hours) the Earth will cover 360 degrees or 2π radians.

A)

In every 24 hours the earth rotates through 360 degrees ( a complete rotation). We are required to determine the length of time it will take the Earth to rotate through 225 degrees. Let x be the duration it takes the earth to rotate through 225 degrees, then the following proportions hold;

(24/360) = (x/225)

solving for x;

x = (24/360) * 225 = 15 hours

B)

In 24 hours the earth rotates through an angle of 2π radians (a complete rotation) . We are required to determine the length of time it will take the Earth to rotate through 9π radians. Let x be the duration it takes the earth to rotate through 9π radians, then the following proportions hold;

(24/2π radians) = (x/9π radians)

Solving for x;

x = (24/2π radians)*9π radians = 108 hours

C)

If the diameter of the earth is 7920 miles, then in a day or 24 hours a point on the equator will rotate through a distance equal to the circumference of the Earth. Using the formula for the circumference of a circle;

circumference = 2*π*R = π*D

                         = 7920*3.142

                         = 24891.43 miles

Therefore a point on the equator covers a distance of 24891.43 miles in 24 hours. This will imply that the speed of the earth is approximately;

(24891.43miles)/(24 hours) = 1037.14 miles/hr

The distance covered by the point in 2 hours will thus be;

1037.14 * 2 = 2074.29 miles

Answer:

2 nickels 3 dimes 5 quarters

Step-by-step explanation:

We know it cannot be the second option because there are only 7 coins and Kim has 10. It also cannot be the first option because the coins add up to $1.60, and Kim has $1.65. It could not be the last option either because the coins add up to $1.15, and Kim has $1.65.

Answer: Third option

Why? The third option has a total of $1.65, which is what Kim has. There are 10 coins all together, which is also what Kim has. And the number of nickels and dimes combined (5) is equal to the number of quarters (5).

Answer:

262 m³

Step-by-step explanation:

The volume (V) of a cone is calculated as

V = [tex]\frac{1}{3}[/tex] πr²h ( r is the radius of the base and h the height )

here diameter = 10 m, hence radius = 5 m

V = [tex]\frac{1}{3}[/tex]π × 5² × 10

   = [tex]\frac{1}{3}[/tex] π × 250 = [tex]\frac{250\pi }{3}[/tex] ≈ 262 m³

Answer:

Option C.(–∞, 2] ∪ (8, ∞)

Step-by-step explanation:

we know that

The solution of the inequality [tex]x > 8[/tex] Is the interval (8,∞)

The solution of the inequality [tex]x\leq 2[/tex] Is the interval (-∞,2]

therefore

The solution of  [tex]x > 8[/tex] or [tex]x\leq 2[/tex] is equal to

(-∞,2] U (8,∞)

Final answer:

The correct interval notation for the inequality x > 8 or x ≤ 2 is (−∞, 2] ∪ (8, ∞), which represents all numbers less than or equal to 2 and all numbers greater than 8.

Explanation:

The question asks to describe the set of numbers using interval notation for the inequality x > 8 or x ≤ 2. Interval notation is a concise way of writing sets of numbers, using brackets to include endpoints and parentheses for exclusive limits. For an inequality like x > 8, we use the notation (8, ∞) to indicate all numbers greater than 8 but not including 8 itself. Likewise, for x ≤ 2, the interval notation is [−∞, 2] which includes all numbers less than or equal to 2. When we combine these with the union because of the 'or' condition, the correct interval notation is (−∞, 2] ∪ (8, ∞). Therefore, the correct answer is C. (8, ∞) indicates all the numbers larger than 8, and (−∞, 2] includes all the numbers up to and including 2.

Answer:

399.6 miles²

Step-by-step explanation:

The area (A) of the major sector is

A = area of circle × fraction of circle

   = πr² × [tex]\frac{255}{360}[/tex]

   = π × 13.4² × [tex]\frac{255}{360}[/tex]

   = [tex]\frac{13.4^2(255)\pi }{360}[/tex] ≈ 399.6 miles²

Answer:

step one: product of powers

step 2:power of powers

step 3: quotient of powers

Step-by-step explanation:

The rule of exponents was applied in each step will be multiplication rule,division rule and inverse of exponent rule.

Arithmetic is a branch of mathematics that studies numbers and the many operations that may be applied to them. The four basic math operations are addition, subtraction, multiplication, and division.

The given expression is;

[tex]\frac{(2)^3(2)^4}{2^{10}}[/tex]

Step 1; Multiplication rule of exponents;

[tex]\frac{(2)^3(2)^4}{2^{10}} \\\\ \frac{2^{3+4}}{2^{10}} \\\\\frac{2^7}{2^{10}}[/tex]

Step 2;Division rule of exponents;

[tex]\frac{2^7}{2{10}} \\\\ (2)^{7-10}\\\\ (2)^{-3}[/tex]

Step 2;Inverse rule of exponents;

[tex](2)^{-3}\\\\ \frac{1}{2^3} \\\\\frac{1}{8}[/tex]

Hence, the rule of exponents was applied in each step will be multiplication rule,division rule and inverse of exponent rule.

To learn more about the arithmetic operations, refer:

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Answer:

  1

Step-by-step explanation:

"Complementary events" by definition have probabilities that total 1. An event is complementary to another if it occurs when the other one doesn't, and vice versa. That is, the outcome is always one or the other of the complementary outcomes--never both, never neither.

The sum of the probabilities of two complementary events is equal to 1.

In probability theory, mutually exclusive events are events that cannot occur simultaneously. The sum of the probabilities of two complementary events is equal to 1. If events A and B are mutually exclusive, then the probability that at least one occurs (A or B) is equal to the sum of their individual probabilities (PA + PB).

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It would have to be 3/5

Answer:

We use the combination formula:

combinations = n! / r! * (n-r)!

n = 5 and r =3

combinations = 5! / 3! * 2!

combinations = 5! / 12

combinations = 5 * 4 * 3 * 2 * 1 / 12

combinations = 5 * 2 = 10

Step-by-step explanation:

Answer:

A. 1 rational root and 2 complex roots

Step-by-step explanation:

The equation [tex]x^3+x^2=x-1[/tex] has the expression [tex]x^3+x^2[/tex] in the left side and the expression [tex]x-1[/tex] in the right side.

when ypu plot the graphs of the functions [tex]y=x^3+x^2[/tex] and [tex]y=x-1,[/tex] the number of intersection points shows the number of real solutions. These two graphs intersect only once, so the equation has one real solution. The third power equation should have three solutions, thus, two remaining solutions are complex.

Answer:

The answer is A

Step-by-step explanation:

The answer is D 3/5

if you can pick 4 and odd it will be 1,3,4,5,7,9

that is 6 numbers, 3/5 5*2 is 10 so 3*2 is 6

3/5