An ancient artifact was recently discovered, but due to rust and corrosion, only 75 grams of the original item remained. Based on historical dates, scientists believe that this artifact was decaying at a rate of 2% each year. Although the artifact will now be preserved at a museum, scientists wonder: how much of the original artifact would there be if they had not discovered it for another 10 years?

Write an exponential function rule and solve. Round your answer to the nearest whole number (the ones place). Enter both the number and the associated units

Answer:

Positive real numbers

Step-by-step explanation:

The domain is all the possible values of x.  Here, x represents the time that the ball falls.

Natural numbers are integers greater than 0 (1, 2, 3, etc.).  However, the time doesn't have to be an integer (for example, x=1.5).

Positive integers are the same as natural numbers.

Positive rational numbers are numbers greater than 0 that can be written as a ratio of integers (1/1, 3/2, 2/1, etc.).  However, the time can also be irrational (for example, x=√2).

Positive real numbers are numbers greater than 0 and not imaginary (don't contain √-1).  This is the correct domain of the function.

The domain for the function ƒ(x) = 1,000 - 16x^2 is Positive Real Numbers.

The domain for the function ƒ(x) = 1,000 - 16x^2 is a set of Positive Real Numbers. This is because in real-world situations, such as the height of an object, negative time values do not make sense, and the function itself involves squaring x, which ensures that the result is positive. Therefore, the appropriate set of numbers as the domain for this function would be Positive Real Numbers.

Answer:

* The shorter side is 270 feet

* The longer side is 540 feet

* The greatest possible area is 145800 feet²

Step-by-step explanation:

* Lets explain how to solve the problem

- There are 1080 feet of fencing to fence a rectangular garden

- One side of the garden is bounded by a river so it doesn't need

 any fencing

- Consider that the width of the rectangular garden is x and its length

 is y and one of the two lengths is bounded by the river

- The length of the fence = 2 width + length

∵ The width = x and the length = y

∴ The length of the fence = 2x + y

- The length of the fence = 1080 feet

∴ 2x + y = 1080

- Lets find y in terms of x

∵ 2x + y = 1080 ⇒ subtract 2x from both sides

∴ y = 1080 - 2x ⇒ (1)

- The area of the garden = Length × width

∴ The area of the garden is A = xy

- To find the greatest area we will differentiate the area of the garden

  with respect to x and equate the differentiation by zero to find the

  value of x which makes the area greatest

∵ A = xy

- Use equation (1) to substitute y by x

∵ y = 1080 -2x

∴ A = x(1080 - 2x)

∴ A = 1080x - 2x²

# Remember

- If y = ax^n, then dy/dx = a(n) x^(n-1)

- If y = ax, then dy/dx = a (because x^0 = 1)

∵ A = 1080x - 2x²

∴ dA/dx = 1080 - 2(2)x

∴ dA/dx = 1080 - 4x

- To find x equate dA/dx by 0

∴ 1080 - 4x = 0 ⇒ add 4x to both sides

∴ 1080 = 4x ⇒ divide both sides by 4

∴ x = 270

- Substitute the value of x in equation (1) to find the value of y

∵ y = 1080 - 2x

∴ y = 1080 - 2(270) = 1080 - 540 = 540

∴ y = 540

* The shorter side is 270 feet

* The longer side is 540 feet

∵ The area of the garden is A = xy

∴ The greatest area is A = 270 × 540 = 145800 feet²

* The greatest possible area is 145800 feet²

Answer:

The height of the tree is Greater than 50 feet, so Iva’s house is not safe

Step-by-step explanation:

step 1

Find the height of the tree

Let

h ----> the height of the tree

we know that

The tangent of angle of 50 degrees is equal to divide the opposite side to the angle of 50 degrees (height of the tree) by the adjacent side to the angle of 50 degrees (tree’s distance from the house)

so

tan(50°)=h/50

h=(50)tan(50°)=59.6 ft

therefore

The height of the tree is Greater than 50 feet, so Iva’s house is not safe

Answer:

greater than and not safe

Step-by-step explanation:

Answer:

The correct answer option is C. [tex] g ( x ) = | x - 4 | + 6 [/tex].

Step-by-step explanation:

We know that the transformation which shifts a function along the horizontal x axis is given by [tex]f(x+a)[/tex], while its [tex]f(x-a)[/tex] which shifts the function to the right side.

Here we are to shift the function 4 units to the right and 6 units up.

Therefore, the function will be:

[tex] g ( x ) = | x - 4 | + 6 [/tex]

Answer: Option C

[tex]g (x) = | x-4 | +6[/tex]

Step-by-step explanation:

If we have a main function and perform a transformation of the form

[tex]g (x) = f (x + h)[/tex]

So:

If [tex]h> 0[/tex] the graph of the function g(x) will be equal to the graph of f(x) displaced h units to the left

If [tex]h <0[/tex] the graph of the function g(x) will be equal to the graph of f(x) displaced h units to the right

Also if the transformation is done

[tex]g (x) = f(x) + k[/tex]

So

If [tex]k> 0[/tex] the graph of the function g(x) will be equal to the graph of f(x) displaced k units up

If [tex]k <0[/tex] the graph of the function g(x) will be equal to the graph of f(x) displaced k units downwards.

In this case the main function is [tex]f(x) = | x |[/tex] and moves 4 units to the right and 6 units to the top, then the transformation is:

[tex]g (x) = f (x-4) +6[/tex]

[tex]g (x) = | x-4 | +6[/tex]

Answer:

Multiply both sides of the equation by 1/3

Step-by-step explanation:

3x^2+18x=21

The first step is to get the x term with a coefficient of 1, so divide by 3 on both sides of the equation

This is the same as multiplying by 1/3

1/3 * (3x^2+18x)=21*1/3

x^2 +6x = 7

Answer:

ok seems easy *ahem*

x=26

Step-by-step explanation:

*MLG intensufys*

Answer:

12.9

Step-by-step explanation:

!!!!

The length of AC is approximately 12.9cm

Given the following identity

Using the SOH CAH TOA identity

sin 40 = AC/20

AC = 20sin40

AC = 20(0.6428)

AC = 12.9cm

Hence the length of AC is approximately 12.9cm

Learn more on SOH CAH TOA here;

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Answer:

B is a function.

Step-by-step explanation:

A relation that is a set of points is a function if all the x's are different.  Each ordered pair in the set (excluding the duplicates if any) all have to have a distinct x.

For example if someone ask you if this is a function they are trying to trick you:

{(4,5),(1,3),(4,5)}.

They are trying to trick you because they listed the element (4,5) twice.

The set is really {(4,5),(1,3)}.

This would be a function because all the x's are distinct, ex 4 is different than 1.

Let's look at your choices:

A This is not a function because you have two pairs with the same x-coordinate.

B This is a function.  There are no repeats of any x,

C This is not a function because you have 11 as x twice.

D This is not a function because x=3 happens twice.

If the same x happens more than once than it isn't a function.  

Answer:

b

Step-by-step explanation:

took test

Answer:

1.B

first picture is x greater than or equal to 4

second picture is x less than or equal to 4

then x>4 is a open circle and a line to the  right

then x<4 to the left

Step-by-step explanation:

Answer:

The approximate volume of the sphere= 4187  cubic units

Step-by-step explanation:

Points to remember

Volume of sphere = (4/3)πr³

Where 'r' radius of sphere

To find the volume of sphere

It is given that the radius of the sphere is 10 units. ,

Here radius r = 10 units

Volume =  (4/3)πr³

 =  (4/3) * 3.14 * 10³

 =  (4/3) * 3140

 =  4186.666 ≈  4187  cubic units

Answer:

Option C.

Step-by-step explanation:

Radius of the sphere is given as 10 units.

We have to calculate the volume of this sphere.

Since formula to measure the volume of sphere is V = [tex]\frac{4}{3}\pi r^{3}[/tex]

V =  [tex]\frac{4}{3}\pi 10^{3}[/tex]

  =  [tex]\frac{4}{3}(3.14)(10^{3})[/tex]

  = 4186.67

  ≈ 4187 cubic units.

Option C. is the answer.

Answer:

The factors are the binomials (x - 7)(x + 2)

Step-by-step explanation:

* Lets explain how to factor a trinomial

- The trinomial  ax² ± bx ± c has two factors (x ± h)(x ± k), where

# h + k = -b/a

# h × k = c/a

- The signs of the brackets depends on the sign of c at first then

  the sign of b

# If c is positive, then the two brackets have the same sign

# If b is positive , then the signs of the brackets are (+)

# If b is negative then the sign of the brackets are (-)

# If c is negative , then the brackets have different signs

* Lets solve the problem

∵ The trinomial is x² - 5x - 14

∴ a = 1 , b = -5 and c = -14

∵ c is negative

∴ The brackets have different signs

∴ (x - h) (x + k) are the factors of the trinomial

∵ h + k = -5/1

∴ h + k = -5 ⇒ (1)

∵ h × k = -14/1

∴ h × k = -14 ⇒ (2)

- From (1) , (2) we search about two numbers their product is 14 and

 their difference is 5 , they will be 7 and 2

∵ 7 × 2 = 14

∵ 7 - 2 = 5

- The sign of b is negative then we will put the greatest number in the

 bracket of (-)

∴ h = 7 and k = 2

∴ The brackets are (x - 7)(x + 2)

* The factors are the binomials (x - 7)(x + 2)

Answer:

a=11.8,b=12,c=8,A=68.7°, B=72°, C= 39.3°

Step-by-step explanation:

Given data:

b = 12

c= 8

a= ?

∠B= 72°

∠C= ?

∠A=?

To find the missing angle we will use law of sine:

a/sinA=b/sinB=c/sinC

Find m∠C.

b/sinB = c/sinC

Substitute the values:

12/sin72°=8/sinC

Apply cross multiplication.

12*sinC=sin72° * 8

sinC=0.951*8/12

sinC=7.608/12

sinC= 0.634

C= 39.3°

Now we know that the sum of angles = 180°

So,

m∠A+m∠B+m∠C=180°

m∠A+72°+39.3°=180°

m∠A=180°-72°-39.3°

m∠A= 68.7°

Now find the side a:

a/sinA=b/sinB

a/sin68.7°=12/sin72°

Apply cross multiplication:

a*sin72°=12*sin68.7°

a*0.951=12*0.931

a=0.931*12/0.951

a=11.172/0.951

a=11.75

a=11.8 ....

Answer:

5.73cm

Step-by-step explanation:

Given parameters:

Circumference of the circle = 36cm

Unknown

Length of the radius = ?

Lets represent the radius by r

Solution

The circumference of a circle is the defined as the perimeter of a circle. The formula is given as:

            Circumference of a circle = 2πr

Since the unknown is r, we make it the subject of the formula:

           r = [tex]\frac{circumference of the circle}{2π}[/tex]

          r = [tex]\frac{36}{2 x 3.142}[/tex] = [tex]\frac{36}{6.284}[/tex] = 5.73cm

Answer:

The radius = 18 cm.

Step-by-step explanation:

I am assuming that is 36π.

If so, then  circumference  = 2 π r and:

2 π r =  36 π

r  = 36π / 2π

r = 18.

For this case we must indicate the sign corresponding to:

[tex]\frac {6} {10}[/tex]and [tex]\frac {9} {12}[/tex]

We have to:

[tex]\frac {6} {10} = 0.6\\\frac {9} {12} = 0.75[/tex]

It is observed that[tex]0.75> 0.6[/tex]

So we have to:

[tex]\frac {6} {10} <\frac {9} {12}[/tex]

Answer:

[tex]\frac {6} {10} <\frac {9} {12}[/tex]

Answer:

After month 3.

Step-by-step explanation:

If you know that the crop will be completely infected after 6 months and the area infected doubles each month, you can work backwards. So picture the 6th month as 100%. Then, basically divide that percentage by 2 until you reach 1/8, or .125, or 12.5. So 100% (month 6) > 50% (month 5) > 25% (month 4) > 12.5% (month 3). So 12.5% is equal to 1/8 or .125, which is what you are trying to look for.

Hope this helps!

The non-food crop will be infected by one-eighth of the crop after 3 months.

The sum of n terms of a geometric sequence is given by the formula,

Sn = [a(1 - r^n)]/(1 - r)

Where r - a common ratio

a - first term

n - nth term

Sn - the sum of n terms

Given that,

The crop is infected by pests.

The area infected doubles after each month. So, it forms a geometric progression or sequence.

The entire crop is infected after six months.

So, n = 6, r = 2(double) and consider a = x

Then the area of the infected crop after 6 months is,

S(6) = [x(1 - 2^6)]/(1 - 2)

      = x(1 - 64)/(-1)

     = -x(-63)

     = 63x

So, after six months the area of the infected crop is about 63x

So, the one-eighth of the crop = 63x/8 = 7.875x

For one month the infected area = x (< one-eighth)

For two months it will be x + 2x = 3x (< one-eighth)

For three months it will be x + 2x + 4x = 7x ( < one-eighth)

For four months it will be x + 2x + 4x + 8x = 15x (> one-eighth)

So, the one-eighth of the crop will be infected after 3 months.

Learn more about the sum of geometric sequences here:

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#SPJ2

Answer:

Step-by-step explanation:

Let CD = x

(x + 40)/2 = 31                  The midline is 1/2 the sum of the 2 bases. multiply by 2 on both sides.

x + 40 = 31 * 2

x + 40 = 62                       Subtract 40 from both sides.

x +40-40 = 62 - 40          Combine

x = 22

Check

(22 + 40)/2

62 / 2

31 = PQ

To find the missing length of a trapezoid with similar triangles inside, we use the ratio of the sides of the triangles, which is approximately 8.667 based on the provided lengths.

When determining the missing length of a trapezoid, we need to consider the properties of similar triangles or the geometric shape of a trapezoid itself. The given information suggests a scenario where similar triangles are within a trapezoid and leads to a proportional relationship between the sides of the triangles. If a trapezoid has triangles within it that share an angle, the lengths of the corresponding sides of those triangles will be proportional.

Based on the data provided, if the long sides of the triangles are in the ratio of 13.0 in to 1.5 in, which simplifies to an approximate factor of 8.667, the bottom sides of the triangles - which are the parallel sides of the trapezoid - will also be in the same ratio. By finding the length of the shorter bottom side, we can divide the length of the longer bottom side by 8.667 to get the missing length on the shorter side.

Answer:

x = 12

Step 1 helps solve the equation, for it helps isolate the x, which is what you are solving for.

Step-by-step explanation:

Isolate the variable x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS:

Step 1: Add 18 to both sides.

6x - 18 (+18) = 54 (+18)

6x = 54 + 18

6x = 72

Step 2: Divide 6 from both sides:

(6x)/6 = (72)/6

x = 72/6

x = 12

x = 12 is your answer.

~

Answer:

[tex]\huge \boxed{X=12}[/tex]

Step-by-step explanation:

Add by 18 from both sides of equation.

[tex]\displaystyle 6x-18+18=54+18[/tex]

Add numbers from left to right.

[tex]\displaystyle 54+18=72[/tex]

[tex]\displaystyle 6x=72[/tex]

Divide by 6 from both sides of equation.

[tex]\displaystyle \frac{6x}{6}=\frac{72}{6}[/tex]

Simplify, to find the answer.

[tex]\displaystyle 72\div6=12[/tex]

[tex]\huge \boxed{x=12}[/tex], which is our answer.

Answer:

[tex]\displaystyle =9x-5[/tex]

Step-by-step explanation:

[tex]\displaystyle 4x-6+5x+1[/tex]

Group like terms.

[tex]\displaystyle 4x-6+5x+1[/tex]

Add the numbers from left to right.

[tex]4+5=9[/tex]

[tex]9x-6+1[/tex]

Add and subtract numbers from left to right to find the answer.

[tex]6-1=5[/tex]

It change to postive to negative sign.

[tex]\displaystyle=9x-5[/tex], which is our answer.

Answer:

The simplified form is 9x - 5

Step-by-step explanation:

It is given an expression in variable x

(4x − 6) + (5x + 1).

To find the simplified form

(4x − 6) + (5x + 1). =  4x − 6 + 5x + 1

 = 4x + 5x - 6 + 1

 = 9x -5

Therefore simplified form of given expression  (4x − 6) + (5x + 1) is,

9x - 5

Answer:

The correct answer is third option.  994

Step-by-step explanation:

Points to remember

Volume of cylinder = πr²h

Where 'r' is the radius and 'h' is the height of cylinder

From the given question we get the cylinder height and radius

The height h = 3 times the diameter of one ball

 = 3 * 7.5 = 22.5 cm

Radius = half of the diameter of a ball

 = 7.5/2 = 3.75 cm

To find the volume of cylinder

Volume of cylinder = πr²h

 = 3.14 * 3.75² * 22.5

 = 993.515 ≈ 994

The correct answer is third option.  994

Answer:

Option B: 81y^4-16x^2, the difference of squares

Step-by-step explanation:

we know that

The Difference of Squares is two terms that are squared and separated by a subtraction sign

so

[tex](a+b)(a-b)=(a^{2}-b^{2})[/tex]

In this problem we have

[tex](9y^{2}-4x)(9y^{2}+4x)[/tex]

Let

[tex]a=9y^{2}[/tex]

[tex]b=4x[/tex]

so

[tex]a^{2}=(9y^{2})^{2}=81y^{4}[/tex]

[tex]b^{2}=(4x)^{2}=16x^{2}[/tex]

substitute

[tex](9y^{2}-4x)(9y^{2}+4x)=81y^{4}-16x^{2}[/tex]

Answer: b

Step-by-step explanation: edge 2022

Answer:

Step-by-step explanation:

[tex]5^{x-2}=625\\\\5^{x-2}=5^4\qquad(5^4=5\cdot5\cdot5\cdot5=625)\\\\5^{x-2}=5^4\Rightarrow x-2=4\qquad\text{add 2 to both sides}\\\\x-2+2=4+2\\\\x=6\\\\\text{check:}\\\\5^{6-2}=5^4=625\qquad\bold{CORRECT}[/tex]

See the attached picture:

Answer:

Answer is option B. 1,111.84 ft²

Step-by-step explanation:

The given dimensions of the un shaded region or rectangle are 18 feet x 22 feet.

Now we have additional 7 feet at both ends to form the diameter of the semicircle, making it [tex]14+18=32[/tex] feet

Radius = [tex]\frac{32}{2}=16[/tex] feet

We have 2 semicircles, one at each end. If we combine it it forms a circle.

Area of the circle (2 semicircles) = [tex]\pi r^{2}[/tex]

= [tex]3.14\times(16)^{2}[/tex] = 803.84 square feet

Now we will find the area of the shaded rectangles above and below the non shaded one. The length is 22 feet and width is 7 feet.

So, area = [tex]22\times7=154[/tex] square feet

We have 2 similar rectangles. So, area of both = [tex]2\times154=308[/tex] square feet

So, total area of shaded region = [tex]803.84+308=1111.84[/tex] square feet.

Answer:

[tex]\large\boxed{a_n=2(-4)^{n-1}=\dfrac{(-4)^n}{2}}[/tex]

Step-by-step explanation:

The explicit equation of a geometric sequence:

[tex]a_n=a_1r^{n-1}[/tex]

The domain is the set of all Counting Numbers.

We have the first term of [tex]a_1=2[/tex] and the second term of [tex]a_2=-8[/tex].

Calculate the common ratio r:

[tex]r=\dfrac{a_{n-1}}{a_2}\to r=\dfrac{a_2}{a_1}[/tex]

Substitute:

[tex]r=\dfrac{-8}{2}=-4[/tex]

[tex]a_n=(2)(-4)^{n-1}\qquad\text{use}\ \dfrac{a^n}{a^m}=a^{n-m}\\\\a_n=(2\!\!\!\!\diagup^1)\left(\dfrac{(-4)^n}{4\!\!\!\!\diagup_2}\right)\\\\a_n=\dfrac{(-4)^n}{2}[/tex]

Answer:

Step-by-step explanation:

A geometric sequence has a common ratio. in this case the common ratio

r = -8/2 = -4.

The explicit formula is an = 2(-4)^(n-1).

Answer:

15) 300%

Step-by-step explanation:

15)

Let the item originally cost n dollars.

The new incorrect price is (n-.6n).

So we want to find k such that (n-.6n)+k(n-.6n)=n+.6n

since we actually wanted it to be (n+.6n).

So we have (n-.6n)+k(n-.6n)=n+.6n

Distribute:

n-.6n+kn-.6nk=n+.6n

Subtract n on both sides

  -.6n+kn-.6nk=.6n

We are trying to solve for k. So add .6n on both sides:

kn-.6nk=1.2n

Divide both sides by n:

k-.6k=1.2

.4k=1.2

Divide both sides by .4

k=1.2/.4

k=3

So 3=300%.

The incorrect price must be increased by 300% to get to the proper new price.

Here is an example:

Something cost $600.

It was reduce by 60% which means it cost 600-.6(600)=600-360=240

This was the wrong price.

We needed it to be increased by 60% which would have been 600+360=960.

So we need to figure out what to increase I wrong price 240 to to get to our right price of 960.

240+k(240)=960

1+k=960/240

1+k=4

k=3

So 240*300%+240 would give me my 960.

16) Speed=distance/time

In the first half hour, she traveled 5 miles (8:30 to 9).

In 1/3 hour she traveled (5-2)=3 miles (9 to 9:20).

We are told not to do anything where she stayed still.

In the last half hour, she traveled (2-0)=2 miles (9:30 to 10).

The average speed=[tex]\frac{5+3+2}{\frac{1}{2}+\frac{1}{3}+\frac{1}{2}}=\frac{10}{\frac{4}{3}}=\frac{10(3)}{4}=\frac{30}{4}=\frac{15}{2}=7\frac{1}{2}[/tex].

Answer:

y = 5x

Step-by-step explanation:

The equation to do this is y = mx + b.

b is y intercept which is 0

m is the slope, which is 5

y = 5x + 0 or y = 5x

Answer:

y=5x

Step-by-step explanation:

So we have the points (0,0) and (2,10) that are on the line.

The slope can be calculate the slope by finding the rise and the run, and then putting the rise/run.

If you want to use the graph to count the rise, you can.  If you start at 0 and need to get to 10, then you need to rise 10 units.

If you want to use the graph to count the run, you can. If you start at 0 and need to get to 2, then you need to run 2 units.

So the slope is 10/2=5.

Or, if you didn't want to count, you could use the slope formula for a line given two points on that line.

That is the formula is [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex].

There is another format for that formula that might be easier to remember.  Just line the points up and subtract vertically. Then put 2nd difference over 1st difference.

So let's do that also:

(  2  ,  10)

-(  0   ,   0)

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

  2           10

So the slope is 10/2=5.

Now you can get the same answer if you had done it the other way:

 ( 0 ,   0)

- ( 2 ,  10)

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

 -2      -10

So the slope is -10/-2=5.

You get the same number either way.

So the slope-intercept form of a line is y=mx+b.

m is the slope and b is the y-intercept.

We found m which is 5.

If you look at the graph, you see the line goes through the y-axis at y=0 so the y-intercept, b, is 0.

The equation of the line here is y=5x+0 or just simply y=5x.

Answer:

See explanation

Step-by-step explanation:

There 2 black shapes (one circle and one square) and 2 white shapes (1 circle and 1 square).

1. The ratio of the number of white shapes to the number of black shapes is 2:2=1:1.

2. The ratio of the number of white circls to the number of black squares is 1:1.

3. The ratio of the number of black circles to the number of black squaress is 1:1.

4. There are 4 shapes and 2 are circles (1 white and 1 black), so circles are [tex]\dfrac{2}{4}=\dfrac{1}{2}[/tex] of all shapes.

a chord intersected by a radius segment at a right-angle, gets bisected into two equal pieces, namely MO = NO and PZ = QZ.

[tex]\bf MO=NO\implies \stackrel{MO}{18}=NO\qquad \qquad NO=6x\implies \stackrel{NO}{18}=6x \\\\\\ \cfrac{18}{6}=x \implies 3=x \\\\[-0.35em] ~\dotfill\\\\ PZ=x+2\implies PZ=3+2\implies PZ=5=QZ \\\\[-0.35em] ~\dotfill\\\\ PQ=PZ+QZ\implies PQ=5+5\implies PQ=10[/tex]

Answer:

the line will pass through yellow and red