8. The table shows the number of different
kinds of sodas sold at a gas station on a
Monday.
A.
B.
Kind of Soda
18
If the gas station had 80 customers on
Tuesday, how many customers can be
predicted to get a Dr. Pepper?
C. 45
Coks
Sprite
Dr. Papper
7-Up
36
Number of
Bottles Sold
11
D. Not Here

Answer:

The median value is 45.

Step-by-step explanation:

"The median is the middle number in a sorted, ascending or descending list of numbers"

The middle number here is 45

50

Step-by-step explanation:

I put the explanation on the attachment. please see it.

The missing lengths of triangles are 5in, 5mi, 13.9km,13.3mi respectively.

What is triangle?

A triangle is a closed, two-dimensional geometric figure with three straight sides and three angles.

What is Pythagorean theorem?

The Pythagorean Theorem is a fundamental theorem in Euclidean geometry that relates to the three sides of a right-angled triangle.

Using the Pythagorean theorem [tex](a^2 + b^2 = c^2)[/tex], we can solve for the missing side in each triangle.

Triangle 1:

[tex]a = 12 in\\\\c = 13 in\\\\a^2 + b^2 = c^2\\\\12^2 + b^2 = 13^2\\\\144 + b^2 = 169\\\\b^2 = 25\\\\b = \sqrt{(25)}\\\\b = 5 in[/tex]

Therefore, the length of the missing side in Triangle 1 is 5 in.

Triangle 2:

[tex]a = 4 mi\\\\b = 3 mi\\\\c = x\\\\a^2 + b^2 = c^2\\\\4^2 + 3^2 = x^2\\\\16 + 9 = x^2\\\\25 = x^2\\\\x = \sqrt{(25)}\\\\x = 5 mi[/tex]

Therefore, the length of the hypotenuse in Triangle 2 is 5 mi.

Triangle 3:

[tex]a = x\\\\b = 11.9 km\\\\c = 14.7 km\\\\a^2 + b^2 = c^2\\\\x^2 + 11.9^2 = 14.7^2\\\\x^2 = 14.7^2 - 11.9^2\\\\x^2 = 192.36\\\\x = \sqrt{(192.36)}\\\\x = 13.9 km[/tex]

Therefore, the length of the height in Triangle 3 is 13.9 km.

Triangle 4:

[tex]a = x\\\\b = 6.3 mi\\\\c = 15.4 mi\\\\a^2 + b^2 = c^2\\\\x^2 + 6.3^2 = 15.4^2\\\\x^2 = 15.4^2 - 6.3^2\\\\x^2 = 178.09\\\\x = \sqrt{(178.09)}\\\\x = 13.3 mi[/tex]

Therefore, the length of the height in Triangle 4 is 13.3 mi.

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The rule that best represents the dilation that was applied to quadrilateral FGHJ to create quadrilateral F'G'H'J' is option B, which is (x, y) à (1.4x, 1.4y).

A dilation is a transformation that changes the size of an object without changing its shape. It is performed by multiplying the coordinates of each point by a scale factor.

In this case, the center of dilation is the origin, which means that the coordinates of each point are multiplied by the same scale factor in both the x and y directions.

The scale factor can be found by comparing the corresponding side lengths of the two quadrilaterals. In this case, the scale factor is 1.4, which means that the lengths of the sides of F'G'H'J' are 1.4 times the lengths of the corresponding sides of FGHJ.

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Using various laws of integers we can say that if a is a positive integer and b is a negative integer, statement A is False, B is True, C s True and D is false.

Here we are given that a is a positive integer while b is a negative integer.

A. The statement says that the difference (a - b) could be negative.

According to the subtraction law of integers, when a negative number is subtracted from a positive number, that is we have

2 - (-3)

Here the 2 minus signs will make a positive to give

2 + 3 = 5

Hence (a - b) will be a positive number since b is negative.

B.

The difference (b - a) cannot be positive.

Since a is positive and b is negative, according to the above example we will get

-3 - 2 = -5

Hence it is true that the difference (b - a) can't be positive.

C.

The sum (a + b) could be positive.

Here, we can see that a is a positive number while b is a negative number. In the light of above example, we will get

2 - 3 = -1

Here the sum is nagative as 3 > 2, but if we had

3 + (-2), then the answer would have been 1. Hence (a + b) can be positive. Hence the statement is true.

D.

The sum (b + a) must be negative.

Integers have commutative properties. Hence a + b = b + a

Hence if a + b can be positive, then b + a can also be positive.

Hence the statement is False.

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To find the extremes of 4x−4y subject to the condition x2 + 2y2 = 1, we can use the method of Lagrange multipliers.

First, we set up the Lagrange equation:

∇f(x,y) = λ∇g(x,y)

where f(x,y) = 4x-4y and g(x,y) = x2 + 2y2 - 1.

Taking partial derivatives, we have:

∂f/∂x = 4
∂f/∂y = -4
∂g/∂x = 2x
∂g/∂y = 4y

Setting these equal to their respective Lagrange multipliers, we have:

4 = 2λx
-4 = 4λy
x2 + 2y2 = 1

Solving for x and y in terms of λ, we get:

x = 2λ/4 = λ/2
y = -λ/4

Substituting these back into the constraint equation, we have:

(λ/2)2 + 2(-λ/4)2 = 1
λ2/4 + λ2/8 = 1
3λ2/8 = 1
λ2 = 8/3

Taking the positive and negative square roots of λ2, we have:

λ = ±2√2/3

Substituting these values back into x and y, we get:

For λ = 2√2/3:
x = (2√2/3)/2 = √2/3
y = -(2√2/3)/4 = -√2/6

For λ = -2√2/3:
x = (-2√2/3)/2 = -√2/3
y = -(-2√2/3)/4 = √2/6

Now we can find the extreme values of f(x,y) by plugging in these values of x and y:

f(√2/3, -√2/6) = 4(√2/3) - 4(-√2/6) = 4√2
f(-√2/3, √2/6) = 4(-√2/3) - 4(√2/6) = -4√2

Therefore, the maximum value of 4x-4y subject to the condition x2 + 2y2 = 1 is 4√2 and the minimum value is -4√2.

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The percent discount on the sneakers is 10%

Casho paid $22.50 before tax, despite the item's $25 tag price. The discount is the difference between the original price and the sale price, which is $25 - $22.50 = $2.50.

The discount is the difference between the original price and the discounted price, expressed as a percentage of the original price.

To find the percent discount, we divide the discount by the original price and multiply by 100:

Percent discount = (discount / tag price) x 100

Percent discount = ($2.50 / $25) x 100

Percent discount = 0.1 x 100

Percent discount = 10%

Therefore, the percent discount on the sneakers is 10%

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Using trigonometry, we can solve for the missing angles to find them as 18.43°, 45°, 45°, and 18.43°.

The sign is mounted on a sloped surface, which means that we'll need to use some trigonometry to find the missing angles.

Let's concentrate on the sign's upper right corner, where the letters x and y are absent from two perspectives. The magnitude of angle x may be determined using trigonometry.

Let's begin by sketching a right triangle that has an angle x. The triangle's two sides may be represented by the sign's vertical and horizontal lines, with the addition of a third side to join the top right corner of the sign to the sloping area below.

Since the sign is an octagon, we know that each interior angle is 135°. Therefore, the measure of angle y must be:

y = 180 - 135 = 45°

Now, let's look at the right triangle that includes angle x. We know that the hypotenuse of the triangle is the sloped surface of the sign, which has a length of 4.5 meters. We also know that the opposite side of the triangle is the height of the sign above the ground, which has a length of 1.5 meters.

Using trigonometry, we can find the measure of angle x by taking the inverse tangent of the opposite side over the adjacent side:

tan(x) = opposite/adjacent = 1.5/4.5 = 1/3

x = tan⁻¹(1/3) ≈ 18.43°

Therefore, the measure of angle x is approximately 18.43 degrees.

Hence, using trigonometry, we can solve for the missing angles to find them as 18.43°, 45°, 45°, and 18.43°.

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The height of the cylinder is 3.6 inches.

We know that the volume of a cylinder of radius R and height H is:

V = pi*R²*H

where pi = 3.14

We know that the radius is R = 5in and the volume is 284.7 inches cubed, replacing that in the formula above we will get:

284.7 in³= 3.14*(5 in)²*H

Solving that for H we will get:

H= (284.7 in³)/ 3.14*(5 in)²

H = 3.6 inches.

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The pole is approximately 3.86 feet tall.

Let's denote the height of the pole as "x" (in feet). From the problem, we know that the pole makes a 75° angle with the road surface, which means that the angle between the pole and the vertical is 90° - 75° = 15°.

Now, we can use the tangent function to find the height of the pole:

tan(15°) = x/15

Multiplying both sides by 15, we get:

x = 15 tan(15°) ≈ 3.86 feet

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by dividing both sides by 1.5.

To solve the problem, we need to first determine the total amount of time Ricky spends in his classes. We know that each class is 1.5 hours long, so if he takes c classes, then he will spend a total of 1.5c hours on class time. In addition, we know that he spends 6.5 hours on homework. Therefore, the total amount of time Ricky spends on his classes and homework is:

Total time = Class time + Homework time

Total time = 1.5c + 6.5

We also know that Ricky has 23 hours per week to dedicate to his classes and homework. Therefore, we can set up the following equation:

Total time = 23

Substituting the expression for a total time from the first equation, we get:

1.5c + 6.5 = 23

Now we can solve for c:

1.5c = 23 - 6.5

1.5c = 16.5

c = 11

Therefore, Ricky takes 11 classes.

The equation that models the question is b) 23=6.5+1.5c. This equation correctly represents the total time Ricky spends on his classes and homework (23 hours), as well as the time he spends on homework (6.5 hours) and the time he spends in class (1.5c hours).

by dividing both sides by 1.5.

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415 square inches of bubble wrap to cover the cylindrical vase that is 16 inches tall with a diameter of 6 inches.

To calculate how much bubble wrap is needed to cover the cylindrical vase, you will need to find the circumference and height of the vase.

First, calculate the circumference of the vase using the diameter of 6 inches:
Circumference = π x diameter
Circumference = 3.14 x 6
Circumference = 18.84 inches

Next, calculate the height of the vase which is given as 16 inches.

To find the surface area of the vase, you will need to multiply the circumference by the height and add the area of the circular bases. The formula for the surface area of a cylinder is:

Surface area = 2πr² + 2πrh
where r is the radius and h is the height.

Since the vase has circular bases, we can find the area of each base by using the formula:
Area of circle = πr²

Now, let's find the radius of the vase:
[tex]Radius = \frac{diameter}{2}[/tex]
[tex]Radius = \frac{6}{2}[/tex]
Radius = 3 inches

So, the area of each base is:

Area of base = π x (radius)²
Area of base = π x 3²
Area of base = 28.27 square inches

The total area of the two bases is 2 x 28.27 = 56.54 square inches.

Now, let's find the surface area of the cylinder:

Surface area = 2πr² + 2πrh
Surface area = 2 x π x 3² + 2 x π x 3 x 16
Surface area = 113.1 + 301.44
Surface area = 414.54 square inches

Therefore, you would need approximately 415 square inches of bubble wrap to cover the cylindrical vase that is 16 inches tall with a diameter of 6 inches.

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The expression that gives the length of the rectangular prism in feet is option D: 27/(2x^2+100).

The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width, and h is the height.

We are given that the volume of the rectangular prism is 135ft^3, and the width is (2x+10)ft. Also, the height is 5 times the width, so h = 5w.

Substituting these values in the formula for the volume, we get:

135 = l(2x+10)(5w)

Dividing both sides by (2x+10)(5w), we get:

l = 135 / (2x+10)(5w)

l = 135 / [10(x+5)w]

Now we can substitute h = 5w:

l = 135 / [10(x+5)h/5]

l = 135 / [2(x+5)h]

l = 135 / [2(x+5)(5w)]

l = 135 / [10(x+5)^2]

Simplifying the expression, we get:

l = 27 / (2(x+5)^2)

Therefore, the expression that gives the length of the rectangular prism in feet is option D: 27/(2x^2+100).

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The cost of each item, obtained from the equation for the sum of the costs of the item are;

A brush costs $3.5

A sketchbook costs $10.5

A paint set costs $21

An equation is a mathematical statement that expresses equivalence between two expression joined by an '=' sign.

The amount Camille brought to the art supply = $39.50

The cost of the brush = (1/3) × The cost of the sketchbook

Cost of the sketchbook = (1/2) × Cost of the paint set

Amunt Camille had left over = $4.50

The cost of the items Camille bought = $39.50 - $4.5 = $35

Let x represent the cost of the brush, let y represent the cost of the sketchbook and let z represent the cost of the paint set

Therefore, we get the following equation; x + y + z = 35

x = (1/3)·y

y = (1/2)·z

Which indicates;

x = (1/3) × (1/2)·z = (1/6)·z

From which we get; (1/6)·z + (1/2)·z + z = 35

(5/3)·z = 3

z = 35 × 3/5 = 21

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Answer:

Step-by-step explanation:

DE/EC.

The volume of the funnel is approximately 42.39 in³. The correct answer is option 2.

To calculate the volume of the funnel, which is shaped like a cone, we need to use the formula for the volume of a cone: V = (1/3)πr²h.

Given:
Height (h) = 4.5 inches
Diameter = 6 inches
Radius (r) = Diameter / 2 = 6 / 2 = 3 inches
Pi (π) ≈ 3.14

Now, plug the values into the formula:

V = (1/3) × 3.14 × 3² × 4.5
V ≈ (1/3) × 3.14 × 9 × 4.5
V ≈ 3.14 × 3 × 4.5
V ≈ 42.39 in³

So, the volume of the funnel is approximately 42.39 in³. The correct answer is option 2.

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The probability that a student chosen randomly from the class plays basketball or baseball is 8/15

Total number of students in Spanish class = 30

Student who plays basketball (A) = 11

Student who plays baseball (B) = 15

Student who plays both sports (A and B) = 10

To find a student who plays basketball or baseball (A or B)

(A or B)  = A + B -  (A and B)

(A or B) = 11 +15 -10

(A or B) = 16

P(A or B) = No. of favorable outcome/ Total no. of outcomes

P(A or B) = 16/30

In simplest form = 8/15

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A ten-by-ten grid has 7 columns shaded. All of the following that represent the part of the grid that is shaded are : The correct answer is (A) 70 and (B) 7.

The information given in the problem tells us that a ten-by-ten grid has 7 columns shaded. Since there are a total of 10 columns in the grid, this means that 7/10 of the columns are shaded.

To express this as a percentage, we can divide 7 by 10 and multiply by 100:

(7/10) x 100 = 70%

Therefore, 70 represents the percentage of columns that are shaded in the grid. Option (A) is correct.

Alternatively, we can express the same proportion as a decimal by dividing 7 by 10:

7/10 = 0.7

Therefore, 0.7 represents the proportion of columns that are shaded in the grid. Option (E) is incorrect because it shows 0.7 as a fraction instead of a decimal.

Option (B) is also correct because it correctly identifies the number of shaded columns as 7. Option (C) is incorrect because it includes both the percentage and the number of shaded columns, which is redundant. Option (D) is incorrect because it shows the proportion of shaded columns as a decimal with an extra zero.

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Answer:

x = 16

Step-by-step explanation:

Multiply the entire first equation by -5 and the entire second equation by 2.

You then get:

15x + 20y = 200

2x - 20y = 72

Add the two equations and you get:

17x = 272

Divide 17 from both sides and you get the answer you need:

x = 16

The expression that shows the total number of pages in Peyton's album is  6 1/2 + f.

We are given that;

Number of pages= 6 1/2

Now,

To write an expression that shows the total number of pages in Peyton’s album, you need to add the number of pages of family photos and the number of pages of friends photos. The expression is:

6 1/2 + f

To evaluate the expression if there are 3 1/2 pages of photos of friends, you need to substitute f with 3 1/2 and then add the fractions. The answer is

6 1/2 + 3 1/2 = 10

Therefore, by the expression the answer will be 6 1/2 + f.

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The containment area is shrinking at a rate of 280π m²/min when the diameter is 80m and the radius is decreasing at a constant rate of 7m/min.

Let's begin by first finding the radius of the containment area when its diameter is 80m.

The diameter of the containment area is 80m, so its radius is half of that:

[tex]r = 80m / 2 = 40m[/tex]

Now, we need to find the rate at which the containment area is shrinking when the radius is decreasing at a constant rate of 7m/min.

We can use the chain rule of differentiation to find this rate:

[tex]dA/dt = dA/dr * dr/dt[/tex]

where A is the area of the containment, t is time, r is the radius of the containment, and dA/dt and dr/dt are the rates of change of A and r with respect to time, respectively.

We know that dr/dt = -7 m/min (negative because the radius is decreasing), and we can find dA/dr by differentiating the formula for the area of a circle with respect to r:

A = π[tex]r^2[/tex]

[tex]dA/dr = 2πr[/tex]

So, when r = 40m, we have:

[tex]dA/dt = dA/dr * dr/dt[/tex]

= (2πr) * (-7)

= -280π [tex]m^2[/tex]/min

Therefore, the containment area is shrinking at a rate of 280π m^2/min when the radius is decreasing at a constant rate of 7m/min and the diameter of the containment area is 80m.

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In a regular tiling, if there are six polygons meeting at a vertex, then the angles at the vertex are 120 degrees.

This is because each regular polygon has interior angles that are multiples of 180 degrees divided by the number of sides. For a regular hexagon, which has six sides, each interior angle measures 120 degrees. When six regular hexagons meet at a vertex in a regular tiling, the total angle sum at the vertex is 720 degrees (6 times 120 degrees).

Since the angles must be divided equally among the six hexagons, each angle at the vertex is 120 degrees.

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Answer:

since 2x = z

replace z with 2x

900000 = x+y+z

900000 = x+y+2x

900000 = 3x+y - eqn (1)

79750= 0.08x +0.09y+0.01z

79750 = 0.08x +0.09y+0.01(2x)

79750 = 0.08x+0.09y+0.02x

79750 = 0.10x +0.09y - eqn(2)

from eqn(1)

900000 = 3x + y

y = 900000-3x - eqn(3)

substitute eqn(3) in eqn(2)

79750 = 0.1x +0.09y

79750=0.1x + 0.09(900000-3x)

79750=0.1x+ 81000 - 0.27x

collect like terms

79750 -81000 = 0.1x-0.27x

-1250 = -0.17x

to find x divide both sides by -0.17

x = -1250/-0.17 ~= 7353

since 2x = z

2*7353 = 14706

in eqn(3)

y = 900000-3x

y= 900000-3(7353)

y = 900000-22059

y = 877941

x =7353,y= 877941,z=14706

The Sample size that is necessary for the selection is =7203

Given that,

[tex]\hat p= 0.25[/tex]

[tex]1 - \hat p = 1 - 0.25 = 0.75[/tex]

margin of error = E = 0.01

At 95% confidence level the z is ,

\alpha = 1 - 95% = 1 - 0.95 = 0.05

[tex]\alpha / 2 = 0.05 / 2 = 0.025[/tex]

Z\alpha/2 = Z0.025 = 1.96 ( Using z table )

Sample size = n = [tex](Z\alpha/2 / E)2 * \hat p * (1 - \hat p)[/tex]

= (1.96 / 0.01)2 * 0.25 * 0.75

= 7203

Sample size =7203

In statistics, the sample size is the measure of the number of individual samples used in an experiment.

The size of the sample holds significant importance in any empirical study that aims to draw conclusions about a larger population based on a smaller sample.

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The height of the flagpole is 26.0 feet.

Height is the vertical distance between two points.

To calculate the height of the flagpole, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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(a) The critical numbers happen when x = 3 or x = -1/2

(b) f is decreasing on (-∞, -1/2), increasing on (-1/2, 3), and increasing on (3, ∞).

(c) f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.

(d) f is concave downward on (-∞, 1/4) and concave upward on (1/4, ∞).

(e) The inflection point of f is at x = 1/4.

(a) To find the critical numbers of f, we need to find the values of x where the derivative of f equals zero or does not exist.

f'(x) = 12x² - 6x - 18 = 6(2x² - x - 3) = 6(x - 3)(2x + 1)

Setting f'(x) equal to zero, we get:

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

x = 3 or x = -1/2

These are the critical numbers of f.

(b) To find the intervals where f is increasing and decreasing, we need to examine the sign of the derivative f'(x) in the intervals determined by the critical numbers.

When x < -1/2, f'(x) < 0, so f is decreasing on the interval (-∞, -1/2).

When -1/2 < x < 3, f'(x) > 0, so f is increasing on the interval (-1/2, 3).

When x > 3, f'(x) > 0, so f is increasing on the interval (3, ∞).

(c) To find the local minimum and maximum values of f, we need to examine the critical numbers and the end points of the intervals.

f(3) = 4(3)³ - 3(3)² - 18(3) + 5 = -22

f(-1/2) = 4(-1/2)³ - 3(-1/2)² - 18(-1/2) + 5 = 25.5

Thus, f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.

(d) To find the intervals where f is concave upward and concave downward, we need to examine the sign of the second derivative f''(x).

f''(x) = 24x - 6 = 6(4x - 1)

When x < 1/4, f''(x) < 0, so f is concave downward on the interval (-∞, 1/4).

1/4 < x, f''(x) > 0, so f is concave upward on the interval (1/4, ∞).

(e) To find the inflection points of f, we need to examine the points where the concavity changes.

The concavity changes at x = 1/4, which is the only inflection point o

In the last 215 days, builders have completed 700 m2 of the alligator habitat that will eventually be 1,200 m2.
It will take approximately 153 days to complete the remaining part of the alligator habitat.

Determine how much longer it will take to complete the alligator habitat, first, we need to find the rate at which the builders are working.
Calculate the work rate
The builders have completed 700 m2 of the 1,200 m2 alligator habitat in 215 days.
Work rate = (completed work) / (number of days)
Work rate = 700 m2 / 215 days = 3.26 m2/day (approximately)
Calculate the remaining work
The total area of the alligator habitat is 1,200 m2, and 700 m2 has been completed.
Remaining work = Total area - Completed work
Remaining work = 1,200 m2 - 700 m2 = 500 m2
Calculate the time to complete the remaining work
Time to complete = (remaining work) / (work rate)
Time to complete = 500 m2 / 3.26 m2/day ≈ 153.37 days
It will take approximately 153 days to complete the remaining part of the alligator habitat.

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It will take approximately 394 more days to complete the alligator habitat.

We can start by finding the proportion of the habitat that has already been completed:

proportion completed = 700 m^2 / 1200 m^2 = 0.5833

This means that there is still 1 - 0.5833 = 0.4167 (or 41.67%) of the habitat left to complete.

Next, we can use a proportion to find out how long it will take to complete the remaining 41.67% of the habitat:

215 days / 0.5833 = x days / 0.4167

Solving for x, we get:

x = 215 days * 0.4167 / 0.5833 ≈ 153 days

Therefore, the total time it will take to complete the alligator habitat is approximately 215 + 153 = 368 days, or about 394 more days from the start.

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You need to add 2125 milliliters of water to fill the square aquarium completely.

We need to find the volume of the square aquarium and then determine the additional water needed to fill it completely. Here are the steps:

1. Convert the given length to meters: 15 cm = 0.15 m
2. Calculate the volume of the square aquarium: Volume = length × width × height. Since it's a square aquarium, all sides are equal, so Volume = 0.15 m × 0.15 m × 0.15 m = 0.003375 cubic meters.
3. Convert the volume to milliliters: 0.003375 cubic meters × 1,000,000 mL/cubic meter = 3375 mL.
4. Calculate the additional water needed: Total volume - Current volume = 3375 mL - 1250 mL = 2125 mL.

You need to add 2125 milliliters of water to fill the square aquarium completely.

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Part A: The equation to represent this situation is: 0.50x + 3 = 18.50
Part B: The equation to represent this situation is: 0.75x + 3 - 0.10x = 21.00
Part C: The equations differ in the following ways:
1. Sydney's equation involves only the price per cup and the tip, while Daria's equation also considers the cost of the plastic cups.
2. The price per cup for Sydney and Daria are different.

Part A: The equation to represent this situation is:

18.50 = 0.50x + 3

Where x represents the number of cups of lemonade sold.

Part B: The equation to represent this situation is:

21.00 = 0.75x + 3 - 0.10x

Where x represents the number of cups of lemonade sold.

Part C: The equations from Part A and Part B differ in that Part B takes into account the cost of each plastic cup used to serve the lemonade, while Part A only considers the revenue from selling each cup of lemonade and the tip received. This means that the profit in Part B is calculated after deducting the cost of each plastic cup from the revenue earned, while the profit in Part A does not account for any costs incurred.

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If a man bought a diamond for $32 in 1682 and the man had instead put the $32 in the bank at 3% interest compounded continuously, then the value of the diamond in 2003 would be $554,311.

The given problem is related to exponential growth. In this problem, the continuous compounding formula will be used to find the value of $32 in 2003.

The formula for continuous compounding is given by:

A = Pert   Where,

P is the principal amount,

r is the annual interest rate,

e is the Euler's number which is approximately 2.71828, and

t is the time in years.

Using the formula, we get:

A = 32e^(0.03 x 321)

A = 32e^9.63

A = 32 x 17322.23

A = $ 554311.36

Thus, $32 invested at 3% compounded continuously from 1682 to 2003 would be worth $554,311.

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