Since Beth was born, the population of her town has increased at a rate of 850 people per year. Beth's 9h birthday, the total population was nearly 307,650. Write and solve a linear equation to find the population on Beth's 16th birthday.

[tex]\underline{\ x|\ \ 0\ \ |\ \ 1\ \ |\ \ 2\ \ |\ \ 3\ \ |\ \ 4\ \ |\ \ 5\ \ |}\\U|500\ |\ 255|\ 130|\ 66\ \ |\ 34\ |\ \ 17\ |\\\\U=a(b)^x\\\\for\ x=0,\ U=500\\\\500=a(b)^0\\\\500=a(1)\to \boxed{a=500}\\\\for\ x=1,\ U=255\\\\255=500(b)^1\\\\255=500b\qquad\text{divide both sides by 500}\\\\b=\dfrac{255}{500}\\\\b=\dfrac{255:5}{500:5}\\\\b=\dfrac{51}{100}\to \boxed{b=0.51}\\\\\text{Therefore we have the equation of the function:}\\\\U=500(0.51)^x[/tex]

[tex]\text{Check for other values of x:}\\\\for\ x=2\\\\U=500(0.51)^2=130.05\approx130\qquad CORRECT\\\\for\ x=3\\\\U=500(0.51)^3=66.3255\approx66\qquad CORRECT\\\\for\ x=4\\\\U=500(0.51)^4=33.826\approx34\qquad CORRECT\\\\for\ x=5\\\\U=500(0.51)^5=17.25125\approx17\qquad CORRECT[/tex]

[tex]Answer:\ \boxed{U=500(0.51)^x}[/tex]

Answer:

92 degrees is x

88 degrees is y

Answer:

x=92.  y= 88

Step-by-step explanation:

all three corners of a tri.  equal 180

so 39+49=88

180-88=92 now y and x= 180 also so y

Answer:

Step-by-step explanation:

Let the length of the rectangular prism = x inches

Width of the rectangular prism = 2 x length = 2x inches

Height of the rectangular prism = 3 inches less than the length = (x -3) inches

Volume of the rectangular prism = length x width x height = 392 cubic inches

= (2x) inches x (x) inches x (x -3) inches = 392 cubic inches

= x2(x-3) = 196 cubic inch

X = 8.59 inch

Length of the rectangular prism = x inches = 7 inch

Width of the rectangular prism = 2 x length = 2 x 7 inch = 14 inch

Height of the rectangular prism = 3 inches less than the length = (x -3) inches =  7 – 3 = 4 inch

The dimensions of the storage container are 7 inches in length, 14 inches in width, and 4 inches in height. The volume equation is solved for length by substituting the relationships between height, width, and length into the formula for volume.

To find the dimensions of a rectangular prism storage container with a volume of 392 cubic inches, we need to set up equations based on the information given. The height (h) of the container is 3 inches less than its length (l), so h = l - 3. The width (w) of the container is twice the length of the container, so w = 2l. Knowing that volume = length * width * height (V = lwh), we can substitute the expressions for h and w into the volume equation to obtain an equation with one unknown:

V = l * (2l) * (l - 3)

Substituting the known volume into the equation, we get:

392 = l * (2l) * (l - 3)

This is a cubic equation that can be solved for l (the length of the container). Once l is found, we can also find h and w since they are defined in terms of l.

Let's solve the equation:

[tex]392 = 2l^2 * (l - 3)[/tex]

Divide both sides by 2 to simplify:

[tex]196 = l^2 * (l - 3)[/tex]

Now we expand and solve for l:

[tex]196 = l^3 - 3l^2[/tex]

Moving all terms to one side gives:

[tex]l^3 - 3l^2 - 196 = 0[/tex]

By trial and error or using a cubic equation solver, we find that l = 7 inches. Now we can find the height and width:

h = l - 3 = 7 - 3 = 4 inches

w = 2l = 2 x 7 = 14 inches

Therefore, the dimensions of the storage container are 7 inches in length, 14 inches in width, and 4 inches in height.

Answer:

(a-f)/6 = r

Step-by-step explanation:

The total Bonnie must pay is the weekend fee plus the hourly rate times the hours worked

Cost = weekend fee * hourly rate* hours

hours = 6

weekend fee =f

hourly rate = r

Cost = a dollars

Substituting in what we know

a = f+ 6r

We want to solve for r

Subtract f from each side

a-f =f-f +6r

a-f = 6r

Divide each side by 6

(a-f)/6 = 6r/6

(a-f)/6 = r

Answer:

-4 is the simplified form. of the given expression.

Step-by-step explanation:

We have been given an expression:

[tex]-64^\frac{1}{3}[/tex]

We can rewrite 64 as [tex]4^3[/tex]

the given expression will be rewritten as:

[tex](-(4)^3)^\frac{1}{3}[/tex]

powers will get cancel out we get:

[tex]-4[/tex] is the required simplified form.

The side length of the equilateral triangle would be 6√3. Therefore, the perimeter would be 18√3 and the area would be 27√3.

Area is derived from side length multiplied by height, which is 9, then dividing by 2.

Answer:

[tex]13\frac{7}{16}[/tex] laps can Fred run in one hour.

Step-by-step explanation:

Given Statement: Fred is running on the school track. He can run [tex]10\frac{3}{4}[/tex]  laps in [tex]\frac{4}{5}[/tex]  of an hour.

Unit rate are expressed as a quantity of 1, such as 3 feet per second or 7 miles per hour, they are called unit rates

From the given condition we have;

In [tex]\frac{4}{5}[/tex] he can run [tex]\frac{43}{4}[/tex] laps

Unit rate per hour = [tex]\frac{\frac{43}{4} }{\frac{4}{5} }[/tex]

                              =[tex]\frac{43}{4} \times \frac{5}{4} = \frac{215}{16}[/tex] laps

Therefore, Fred can run in one hour is, [tex]13\frac{7}{16}[/tex] laps.

Final answer:

To determine how many laps Fred can run in one hour, we used a proportion equation, converting 10 3/4 laps to an improper fraction and then solving for x. Fred can run approximately 13.44 laps in one hour.

Explanation:

Calculating Laps per Hour

To find out how many laps Fred can run in one hour, we need to perform a simple proportion based on the information given. Fred can run 10 3/4 laps in 4/5 of an hour. To find out how many laps he can run in a full hour (1 hour), we set up the proportion equation:

\[\left(\frac{10 \frac{3}{4}}{\frac{4}{5}}\right) = \left(\frac{x}{1}\right)\]

First, we convert 10 3/4 to an improper fraction, which is \(\frac{43}{4}\). Then we solve for \(x\) by multiplying both sides by 1, which simplifies our proportion to:

\[x = \frac{43}{4} \times \frac{5}{4}\]

Next, we multiply the numerators and then the denominators:

\[x = \frac{43 \times 5}{4 \times 4}\]

\[x = \frac{215}{16}\]

Finally, we divide 215 by 16 to get \(x = 13.4375\), which means Fred can run approximately 13.44 laps in one hour.

Step-by-step explanation:

We have been given that AE=BE and [tex]\angle1\cong \angle2[/tex].

We can see that angle CEA is vertical angle of angle DEB, therefore, [tex]m\angle CEA=m\angle DEB[/tex] as vertical angles are congruent.

We can see in triangles CEA and DEF that two angles and included sides are congruent.

[tex]\angle 1\cong \angle 2[/tex]

[tex]AE=BE[/tex]

[tex]\angle CEA\cong\angle DEB[/tex] or [tex]\angle 3\cong \angle 4[/tex]  

Therefore, [tex]\Delta CEA\cong \Delta DEB[/tex] by ASA postulate.

Since corresponding parts of congruent triangles are congruent, therefore CE must be congruent to DE.

Set up a proportion:

17 miles in 306 minutes, write as 17/306

X miles in 162 minutes, write as X/162

Now set to equal and solve:

17/306 = x/162

Cross multiply:

17 * 162 = 306 *x

2754 = 306x

Divide both sides by 306:

x = 2754 / 306

x = 9

It will travel 9 nautical miles.

Answer:

B

Step-by-step explanation:

9y² is a perfect square since [tex]\sqrt{9y^2}[/tex] = 3y

10x² is not a perfect square since [tex]\sqrt{10x^2}[/tex] = [tex]\sqrt{10}[/tex] x

Answer:

B is the answer for this equation

Answer:

Length=9 inches,width=4 inches, height =2 inches.

k= [tex]-\frac{111}{8}[/tex]

Step-by-step explanation:

a) If by synethetic division method the remainder equals i.e in the last row and last column if figure yield is 0 then 2 is one of the factor i.e one of the solution of given equation.

Remainder is 0.

Hence 2 is the solution of given equation.

b) Given a box having volume 72 cubic inches

   Let Height = x inches

   ∵ length is 7 inches more than the height

  ⇒     Length = x+7 inches

  & also width is twice the height

 ⇒      Width = [tex]2\times x[/tex]

   Given   Volume = 72 cubic inches

   [tex]length\times width\times height[/tex] = 72

   [tex](x+7)\times (2x)\times x[/tex] - 72 = 0

From part a, 2 is the solution of above equation

⇒       Length = x+7 = 2+7 = 9 inches

          Width = [tex]2\times2[/tex] = 4 inches

         Height = x = 2 inches

Given [tex]4x-3[/tex] is a factor of [tex]20x^3+23x^{2} -10x+k[/tex]

Hence, [tex]20(\frac{3}{4})^3 +23(\frac{3}{4}) ^{2} -10(\frac{3}{4})+k=0[/tex]

            [tex]\frac{135}{16}+\frac{207}{16}-\frac{120}{16} +k=0[/tex]

                   [tex]\frac{111}{8}+k=0[/tex]

                      k= [tex]-\frac{111}{8}[/tex]

Synthetic division confirmed that 2 is a solution to the equation 2h3 +14h2 - 72 = 0. Using this information, we found the dimensions of a box with given characteristics to be: height, 2 inches; width, 4 inches; length, 9 inches. Lastly, used synthetic division again to find k so that 4x-3 is a factor of 20x3+23x2-10x+k.

To solve the first portion of the question, we will use synthetic division to prove that 2 is a solution of the given equation (2h3 +14h2 - 72 = 0). By placing the coefficients into a row (2, 14, 0, -72) and bringing down the first coefficient (2), we multiply by our solution (2), add down the column, and repeat until the end. If we reach 0 at the end, this shows that 2 is indeed a solution.

For the second part of the question, we are given the volume of the box as 72 cubic inches, and we already know from part a that the height, h, of the box is equal to our solution, 2. So, the width of the box would be twice the height, or 2*2 = 4 inches. The length of the box is 7 inches more than the height, or 2+7 = 9 inches.

Lastly, to find the value of k so that 4x-3 is a factor of 20x3+23x2-10x+k, one would set up synthetic division using the roots of 4x-3 and the coefficients of the polynomial, then solve for k when the remainder equals zero.

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To find the number of red fish in a tank where the ratio of red to blue fish is 1:5 and there are 15 blue fish, divide the number of blue fish by the blue part of the ratio (15 ÷ 5) and multiply by the red part (1), resulting in 3 red fish.

The question asks how many red fish are there in a fish tank if the ratio of red fish to blue fish is 1:5 and there are 15 blue fish. To solve this, you can use the ratio provided. Since the ratio of red to blue fish is 1:5, for every 1 red fish, there are 5 blue fish.

Given there are 15 blue fish, you can divide the number of blue fish by the ratio part for blue fish to find out how many parts of red fish there are.

Step 1: Find the ratio part representing red fish. It is 1.

Step 2: Divide the number of blue fish by the blue ratio part (5) to find how many times the ratio fits into the blue fish population. 15 blue fish ÷ 5 = 3.

Step 3: Multiply the result by the red ratio part. 3 × 1 = 3 red fish.

Answer:

Option B is correct.

Weight of  a bowling ball is 5.24 Ib

Step-by-step explanation:

Assume: The shape of the bowling ball is perfectly spherical.  

Given:  

Radius of a bowling ball= 5 inches (r)  .

One cubic inch weighs [tex]\frac{1}{100}[/tex]th of a pound.

Density of a bowling ball = [tex]\frac{1}{100} Ibs/in^3[/tex]

Volume of sphere is given by:

[tex]V = \frac{4}{3} \pi r^3[/tex] where V is the volume and r is the radius of the sphere.

Substitute the value of r =5 and [tex]\pi = 3.14[/tex] in above we get;

[tex]V = \frac{4}{3} \cdot 3.14 \cdot 5^3 =\frac{4}{3} \cdot 3.14 \cdot 125[/tex]

Simplify:

[tex]V = 523.3333... in^3[/tex]

To find the weight of a bowling ball:

[tex]Weight = Volume \times Density[/tex]

Then;

[tex]Weight = 523.33333.. \times \frac{1}{100} =\frac{523.3333..}{100} = 5.2333...[/tex]

Therefore, the weight of a bowling ball ≈ 5.24 Ib

Answer:

The weight of a bowling ball is 5.24 pounds.

Option (B) is correct.

Step-by-step explanation:

Formula

[tex]Volume\ of\ a\ sphere = \frac{4}{3}\pi\ r^{3}[/tex]

Where r is the radius of a sphere.

As given

The radius of the ball is 5 in.

As the shape of the ball is spherical .

Thus

[tex]Volume\ of\ a\ ball = \frac{4}{3}\pi\ 5^{3}[/tex]

[tex]\pi = \frac{22}{7}[/tex]

Thus

[tex]Volume\ of\ a\ ball = \frac{4\times 22\times 5\times\ 5\times 5}{3\times 7}[/tex]

[tex]Volume\ of\ a\ ball = \frac{11000}{21}[/tex]

Volume of a ball = 523.8 in³ (Approx)

As

[tex]1\ in^{3} = \frac{1}{100}\ pound[/tex]

Thus

Convert  523.8 in³ into pounds.

[tex]523.8\ in^{3} = \frac{523.8}{100}\ pound[/tex]

[tex]523.8\ in^{3} = 5.24\ pound\ (Approx)[/tex]

Therefore the weight of a bowling ball is 5.24 pounds.

Therefore Option (B) is correct.

Answer:

d

Step-by-step explanation:

Answer:

Given the vertices of the rectangle ABCD:

A = (1, 7) , B = (5, 3), C = (3,1) and D = (-1, 5)

Distance(D) formula for two points is given by;

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

Using distance formula:

[tex]AB = \sqrt{(5-1)^2+(3-7)^2}=\sqrt{(4)^2+(-4)^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}[/tex] units

[tex]BC= \sqrt{(3-5)^2+(1-3)^2}=\sqrt{(-2)^2+(-2)^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}[/tex] units.

[tex]CD = \sqrt{(-1-3)^2+(5-1)^2}=\sqrt{(-4)^2+(4)^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}[/tex] units

[tex]DA= \sqrt{(-1-1)^2+(7-5)^2}=\sqrt{(-2)^2+(2)^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}[/tex] units.

Since, the Opposite sides of a rectangle are the same length.

⇒AB = CD and BC =DA

Area of rectangle is equal to multiply its width by length.

Area of rectangle ABCD = [tex]CD \times BC[/tex]

= [tex]4\sqrt{2} \times 2\sqrt{2} = 8 \times 2 = 16[/tex] square units.

Therefore, the area of rectangle is, 16 square units.

The answer: C) (-1, -1)

The equation in point slope form is y-y1 = m(x-x1)

The first point given is (-13,9) so this is used for Y1 and X1.

m is the slope which is found by the change in Y over the change in x.

The slope is -3 - 9 / 11 - -13, which equals -1/2

So the equation becomes y-9 = -1/2(x+13)

Answer:

y-9=-1/2(x+13)

Step-by-step explanation:

To find the slope for the line, we use

m = (y2-y1)/(x2-x1)

since we know the points( -13,9) and (11,-3)

m= (-3-9)/(11--13)

   =(-3-9)/(11+13)

  =-12/24

  = -1/2

We can use the point slope form to make the equation for the line

y-y1=m(x-x1)

y-9=-1/2(x--13)

y-9=-1/2(x+13)

Answer:

AC is greater than BC because segment AC is the hypotenuse of right triangle ABC, and the hypotenuse is the longest side of a right triangle.

Answer: -3/5

Step-by-step explanation:

All you have to do is use rise over run!

basically form a triangle and count how much up and how much over. :D

Answer:

[tex]\frac{x^2}{64} + \frac{y^2}{25} =1[/tex]

Step-by-step explanation:

An ellipse with vertices (-8, 0) and (8, 0)

Distance between two vertices = 2a

Distance between (-8,0) and (8,0) = 16

2a= 16

so a= 8

Vertex is (h+a,k)

we know a=8, so vertex is (h+8,k)

Now compare (h+8,k) with vertex (8,0) and find out h and k

h+8 =8, h=0

k =0  

a minor axis of length 10.

Length of minor axis = 2b

2b = 10

so b = 5

General formula for the equation of horizontal ellipse is

[tex]\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b} =1[/tex]

a= 8 , b=5 , h=0,k=0. equation becomes

[tex]\frac{(x-0)^2}{8^2} + \frac{(y-0)^2}{5} =1[/tex]

[tex]\frac{x^2}{64} + \frac{y^2}{25} =1[/tex]

To calculate the height of Lucy's cherry tree, we first convert 3 7/10 to an improper fraction, resulting in 37/10. Then we multiply the height of the lemon tree (6 feet) by 37/10 to get 22.2 feet. Thus, the cherry tree is 22.2 feet tall.

To find the height of Lucy's cherry tree, which is 3 7/10 times as tall as her lemon tree, we start with the known height of the lemon tree, which is 6 feet tall. We then multiply this height by the factor 3 7/10 to determine the height of the cherry tree.

First, let's convert the mixed number to an improper fraction to simplify the calculation:

Next, we'll multiply the height of the lemon tree by this fraction:

6 feet × 37/10 = (6 × 37) / 10 = 222 / 10 = 22.2 feet.
So, the height of Lucy's cherry tree is 22.2 feet tall.

Answer:

Each day, [tex]18[/tex] tours are offered.

Step-by-step explanation:

We were given that, the museum starts at,

10:00 AM and closes at 4:00 PM.

The total duration is [tex]4:00PM-10:00AM=16:00GMT-10:00GMT=6hours[/tex]

Since the tour of the museum are offered every [tex]\frac{1}{3}[/tex] hour each day, we can calculate the number of tours that will be offered within the 6 hours as follows

[tex]Number\:of\:tours=\frac{6}{\frac{1}{3} }[/tex]

We rewrite to obtain,

[tex]Number\:of\:tours=6\div \frac{1}{3}[/tex]

We multiply by the reciprocal of the second fraction to get,

[tex]Number\:of\:tours=6\div \frac{3}{1}=18[/tex]

Therefore 18 tours will be offered each day in the art museum.

Answer: Therefore, the art museum offers 18 tours each day.

Step-by-step explanation: The art museum offers tours every 1/3 hour starting at 10 A.M. and closes at 4:00 P.M. To determine how many tours are offered each day, we need to find the total number of 1/3 hour intervals between 10 A.M. and 4:00 P.M.

First, we need to convert the closing time to the same format as the starting time. Since there are 60 minutes in an hour, we can write 4:00 P.M. as 16:00.

Next, we calculate the number of hours between the starting and closing times by subtracting 10 from 16, which gives us 6 hours.

Since there are 3 intervals in each hour, we multiply the number of hours by 3 to find the total number of 1/3-hour intervals. In this case, 6 hours multiplied by 3 equals 18 intervals.

Therefore, the art museum offers 18 tours each day.

The selling price of the necklace is calculated by adding a 75% markup to the original purchase price of $150, which results in a selling price of $262.50.

To calculate the selling price of a necklace that a jewelry store purchased for $150 with a 75% markup, you first need to find out how much 75% of the purchase price is and then add that to the original purchase price. To do this, multiply the purchase price, $150, by 75% (or 0.75). This calculation gives you the markup amount:

Markup amount = $150 imes 0.75 = $112.50

After finding the markup amount, you add it to the original purchase price to find the selling price:

Selling price = Original purchase price + Markup amount

Selling price = $150 + $112.50 = $262.50

Therefore, the jewelry store will sell the necklace for $262.50.

Answer:

 the required ratio is: 4:9

Step-by-step explanation:

Quinn has 4 cousins who live in Texas, 3 cousins who live in Nebraska, and 9 cousins who live in Michigan.

We have to find the ratio of Quinn's cousins who live in Texas to her cousins who live in Michigan

The required ratio is:

[tex]Ratio=\frac{\text{Number of Quinn's cousins who live in Texas}}{\text{cousins who live in Michigan}}[/tex]

[tex]Ratio=\frac{4}{9}[/tex]

Hence, the required ratio is: 4:9

The ratio of Quinn's cousins who live in Texas to her cousins who live in Michigan is 4:9. This means for every 4 cousins in Texas, there are 9 cousins in Michigan.

The subject of this question is Mathematics. Specifically, it involves calculating ratios. In this case, the student wants to know the ratio of Quinn's cousins who live in Texas to her cousins who live in Michigan.

The ratio of something is simply a way to compare quantities. Here, we have 4 cousins in Texas and 9 cousins in Michigan. So, to get the ratio from Texas to Michigan, we simply write it as '4:9' or we can say, 'for every 4 cousins in Texas, there are 9 cousins in Michigan'.

In summary, the ratio of Quinn's cousins who live in Texas to her cousins who live in Michigan is 4:9.

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Answer:

€9.31

Step-by-step explanation:

The relationship of tax to price is ...

... tax : price = 0.19 : 1

Then the relationship of tax to total cost is ...

... tax : (tax + price) = 0.19 : (0.19 +1) = 0.19 : 1.19 = 19 : 119

So, the tax is 19/119 times the total cost. In this problem, ...

... VAT = 19/119 × €58.31

... VAT = €9.31

Answer:

62 steps per minute.

Step-by-step explanation:

We have been given that Robert climbed 775 steps in [tex]12\frac{1}{2}[/tex] minutes.

To find the the average steps per minute we will divide 775 by [tex]12\frac{1}{2}[/tex].

[tex]\text{The average steps per minute}=775\div 12\frac{1}{2}[/tex]

Let us convert our mixed fraction into improper fraction.

[tex]\text{The average steps per minute}=775\div \frac{25}{2}[/tex]

Dividing a number by a fraction is same as multiplying the number by the reciprocal of fraction.

[tex]\text{The average steps per minute}=775\times \frac{2}{25}[/tex]

[tex]\text{The average steps per minute}=31\times2[/tex]

[tex]\text{The average steps per minute}=62[/tex]

Therefore, Robert climbed 62 steps per minute.

Answer:

62 Steps per minute. <3

Step-by-step explanation: