Creating an Exponential Model
In this activity, you will formulate and solve an exponential equation that models a real-world situation.
Emma doesn't have experience using credit cards. In fact, she just got her first one. She is also about to start her first year
of college. She uses her new credit card to purchase textbooks for her classes. The total comes to $300. These are the
terms of her credit card:
• It has a 15% annual interest rate.
• The interest is compounded monthly
• The card has $0 minimum payments for the first four years it is active.
The expression that models this situation is P(1 + r/n)^nt, where (1 + r/n) represents the growth factor of the interest rate. Emma also wonders how long it will take her balance of $300 to reach $450

Answer:

5x - 15

Step-by-step explanation:

5(x - 3) ← multiply each term in the parenthesis by the 5 outside

= 5x - 15

The lengths of all four sides of parallelogram ABCD:

AB = 15 cm, BC = 33 cm, CD = 15 cm and AD = 33 cm

Consider a parallelogram ABCD

Let's assume that x represents the length of sides AD and BC

y be the length of sides AB and CD

We know that the formula for the perimeter of parallelogram is

P = 2(length + width)

Here,  AD is 3 cm more than twice AB

So we get an expression,

AD = 3 + 2(AB)

AD = 3 + 2y

x  = 3 + 2y

USing above formula of perimeter, the perimeter of parallelogram ABCD would be,

P = 2(x + y)

96 = 2(3 + 2y + y)

48 = 3 + 3y

45 = 3y

y = 15 cm

And x = 3 + 2(15)

x = 33 cm

Therefore, the lengths of sides AD and BC =  33 cm and the lengths of sides AB and CD = 15 cm

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We can construct a shape of area 4 square meters using the four shapes square, rectangle, triangle and parallelogram.

Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape.

The area of a plane figure is the area that its perimeter encloses. The amount of units in a figure determines its area. Area is measured in square units like cm² and m². Area of a shape is a two dimensional quantity.

Area of a square =  side² (a²)

Area of a triangle = 1/2 × base length × height (1/2×b×h)

Area of a rectangle = length × breadth ( l×b )

Area of a parallelogram = base length × height ( b×h)

Consider the square of side 1 m

Area = 1 × 1 = 1 m²

Now consider the triangle of base length , b = 1/2 m and height, h = 2 m

∴ Area = 1/2×1/2×2 = 1/2 m²

Once more , area of the parallelogram of side 1 m and height 2 m is given by

Area = 1×2= 2 m²

For the rectangle of length 1/2 m and breadth 1 m

Area = l × b = 1/2 × 1 = 1/2 m²

∴ Total area = 1 + 1/2 + 2 + 1/2 = 4 square meter.

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The approximate volume of the cooking pot will be;  2009.6 cubic inches.

Since volume of the cylinder is the product of the height, pie, and square of the radius.

The volume of the cylinder = [tex]\pi r^{2} h[/tex]

We know that the volume of the cylinder is  [tex]\pi r^{2} h[/tex]

Here we are given that Trevor is making soup in a cylindrical cooking pot that has a diameter of 10 inches and a height of 8 inches.

h=10 in, and r= 8 in, so

[tex]\begin{gathered} V=(3.14)(10 in)(8 in)^2 \\ V= 2009.6 in^3 \end{gathered}[/tex]

Then, the pot can hold 2009.6 cubic inches

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Answer:

(x-3)^2+(y-2)^2=16

Step-by-step explanation:

The equation of a circle in standard form is (x-h)^2+(y-k)^2=r^2

So, the center is (3,2), so you have to input the opposite value.

(x-3)^2+(y-2)^2=4^2

simplify:
(x-3)^2+(y-2)^2=16

The x - coordinate of B given the y - coordinate, can be found to be 3 units.

Looking at the figure, we notice that the relationship between B and the circle can be modeled as a right triangle. 5 would be the hypotenuse, 4 which is the y - coordinate would the be the height.

The x - coordinate would then be the length which can be found by the Pythagorean theorem.

The x - coordinate is:

= 5² = 4² + x²

Simplifying the equation, we get:

25 = 16 + x²

9 = x²

x = 3

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The height of the candle change over time as a linear function

The height of the candle decreases by 2 inches every hour, which means the height of the candle is a linear function of time.

We can write the equation for the height of the candle as:

h(t) = 12 - 2t

where h is the height of the candle in inches and t is the time in hours.

If the candle burned for 2 hours, we can plug in t=2 into the equation to find the height:

h(2) = 12 - 2(2) = 8

So the height of the candle after 2 hours of burning is 8 inches.

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The total number of possible outcomes when flipping a coin 8 times is 2^8 = 256.

a. To get exactly 6 heads, we need to choose 6 out of the 8 flips to be heads, and the remaining 2 flips to be tails. The number of ways to choose 6 out of 8 is given by the binomial coefficient (8 choose 6) = 28. Therefore, there are 28 possible outcomes with exactly 6 heads.

b. To get at least 3 heads, we need to count the number of outcomes with 3, 4, 5, 6, 7, or 8 heads. Using the same logic as part (a), we can count the number of outcomes with each of these numbers of heads and add them up to get the total number of outcomes with at least 3 heads.

Number of outcomes with 3 heads: (8 choose 3) = 56

Number of outcomes with 4 heads: (8 choose 4) = 70

Number of outcomes with 5 heads: (8 choose 5) = 56

Number of outcomes with 6 heads: (8 choose 6) = 28

Number of outcomes with 7 heads: (8 choose 7) = 8

Number of outcomes with 8 heads: (8 choose 8) = 1

Total number of outcomes with at least 3 heads = 56 + 70 + 56 + 28 + 8 + 1 = 219.

c. To get at most 6 heads, we need to count the number of outcomes with 0, 1, 2, 3, 4, 5, or 6 heads. Using the same logic as part (b), we can count the number of outcomes with each of these numbers of heads and add them up to get the total number of outcomes with at most 6 heads.

Number of outcomes with 0 heads: (8 choose 0) = 1

Number of outcomes with 1 head: (8 choose 1) = 8

Number of outcomes with 2 heads: (8 choose 2) = 28

Number of outcomes with 3 heads: (8 choose 3) = 56

Number of outcomes with 4 heads: (8 choose 4) = 70

Number of outcomes with 5 heads: (8 choose 5) = 56

Number of outcomes with 6 heads: (8 choose 6) = 28

Total number of outcomes with at most 6 heads = 1 + 8 + 28 + 56 + 70 + 56 + 28 = 247.

d. To get at most 2 heads, we need to count the number of outcomes with 0, 1, or 2 heads. Using the same logic as parts (b) and (c), we can count the number of outcomes with each of these numbers of heads and add them up to get the total number of outcomes with at most 2 heads.

Number of outcomes with 0 heads: (8 choose 0) = 1

Number of outcomes with 1 head: (8 choose 1) = 8

Number of outcomes with 2 heads: (8 choose 2) = 28

Total number of outcomes with at most 2 heads = 1 + 8 + 28 = 37.

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A function has access to parameters that are passed to it and any variables created within the function (local and function scope).

The variable names that a function has access to depend on the scope of the variables. Scope refers to the visibility of a variable within a program.

There are four different scopes that a variable can have in relation to a function:
Global scope -

A variable that is declared outside of a function has global scope.

This means that it can be accessed by any function within the program.
Local scope -

A variable that is declared within a function has local scope.

This means that it can only be accessed within the function in which it was declared.
Parameter scope -

A variable that is passed as a parameter to a function has parameter scope.

This means that it can only be accessed within the function in which it was passed.
Function scope -

A variable that is created within a function has function scope.

This means that it can only be accessed within the function in which it was created.

It does not have access to variables declared outside of the function (global scope) unless they are explicitly passed as parameters.

It is important to note that naming conflicts can occur if variable names are not chosen carefully, especially when dealing with global and local scope.

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The statement is true: If x and y are linearly independent, and if z is in Span {x, y}, then {x, y, z} is linearly dependent.

The statement is true.
Let's first understand the terms used:
Linearly independent:

A set of vectors is linearly independent if none of them can be expressed as a linear combination of the other vectors. In other words, no vector in the set can be written as a sum of scalar multiples of the other vectors.
Span:

The span of a set of vectors is the set of all linear combinations of those vectors.

In this case, Span[tex]{x, y}[/tex] is the set of all vectors that can be formed by adding scalar multiples of x and y.
Now, let's consider the given statement:
If x and y are linearly independent, it means that neither x nor y can be expressed as a linear combination of the other. However, it is given that z is in the Span[tex]{x, y}.[/tex]

This means that z can be expressed as a linear combination of x and y:
[tex]z = ax + by[/tex], where a and b are scalar constants.
Let's analyze the set[tex]{x, y, z}[/tex]. We know that z can be expressed as a linear combination of x and y, as shown above. This implies that the set [tex]{x, y, z}[/tex]is linearly dependent, because one vector (z) can be expressed as a linear combination of the others [tex](x and y)[/tex].

Thus, the statement is true: If x and y are linearly independent, and if z is in Span[tex]{x, y}[/tex], then[tex]{x, y, z}[/tex] is linearly dependent.

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Answer:

First image.

Step-by-step explanation:

Second image.

since the formula for the area of a circle squares the radius, the area of the larger circle is always 4 or 22 multiple the tiny square

Answer:

Cube

Step-by-step explanation:

A cube is a three-dimensional shape with six square faces that are all congruent to each other. Each face of a cube is perpendicular to the adjacent faces, and all of its edges are the same length.

Answer: If every hour the candle decreases in height by 2 inches, turn it into a chart! If you learned this already, figure out which variable is the constant and the decrease in height for each hour. Or you could just multiply the amount of hours by 2, there for 2x2=4 so that a 4 inch decrease. 12-4=8 and the candle will be 8 inches tall. Hope this helps :D

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want the measures of x and ∠ECB in rectangle ABCD with center point E and AE=3x+6, BD=5x+19, ∠EBA=25°.

The diagonals of a rectangle are congruent and cross at their midponts. This means the triangles they form are all isosceles.

The value of x can be found by relating the length of a half-diagonal to the length of the whole diagonal.

  2·AE = BD

  2(3x +6) = 5x +19 . . . . . . . substitute given lengths

  6x +12 = 5x +19 . . . . . eliminate parentheses

  x = 7 . . . . . . . . . . . subtract 5x+12

The value of x is 7.

Angle EBC is the complement of angle EBA, so its measure is ...

  ∠EBC = 90° -∠EBA

  ∠EBC = 90° -25° = 65°

Angle ECB is the other base angle of isosceles triangle BCE, so has the same measure as ∠EBC.

  ∠ECB = 65°

__

Additional comment

You know the corner angles of a rectangle are 90°, which is what makes angles EBA and EBC complementary.

With x = 7, AE = 27 and BD = 54. The full diagonal is double the length of the half-diagonal, as you expect.

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The circle graph that correctly represents the data in the table is that with:  A. four sections, labeled turkey 30 percent, ham 20 percent, chicken 35 percent, and vegetarian 15 percent correctly represents the data in the table.

To find the percentage for each type of sandwich, we can divide the number of customers who ordered that sandwich by the total number of customers (100) and multiply by 100.

Using the data given in the table, we get:

Turkey: 30/100 x 100% = 30%

Ham: 20/100 x 100% = 20%

Chicken: 35/100 x 100% = 35%

Vegetarian: 15/100 x 100% = 15%

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Eliminate x by adding the two equations, yielding [tex]y = 6[/tex]. Then, substitute y into either equation to solve for x, obtaining [tex]x = -3[/tex]. The solution is [tex](-3,6)[/tex].

An equation is a mathematical statement that uses symbols, numbers, and variables to show that two expressions are equal, typically with an equal sign between them.

A variable is a quantity or symbol that can take on different values in a mathematical expression, equation, or formula, typically represented by a letter.

To eliminate a variable in a system of equations, we want to find a way to add or subtract the two equations so that one of the variables is eliminated.

In this case, we can eliminate the variable x by adding the two equations

[tex]x- 7y = -45[/tex]

[tex](-x +8y = 51)[/tex]

[tex]y = 6[/tex]

Now that we have found the value of y, we can substitute it into either of the original equations to solve for x. Let's use the first equation

[tex]x-7y = -45\\x- 7(6) = -45\\x = -45+42\\x = -3[/tex]

Therefore the solution to the system of equations is [tex]x= -3[/tex] and[tex]y= 6[/tex]

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the solution to the system of equations is (x, y) = (-3, 6).

What is an Equations?

Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.

To eliminate the variable x, we can add the two equations together. This will result in the x variable being eliminated, leaving us with an equation in terms of y:

(x - 7y) + (-x + 8y) = -45 + 51

Simplifying and solving for y, we get:

y = 6

Now that we know the value of y, we can substitute it into one of the original equations to solve for x. Using the first equation:

x - 7(6) = -45

Simplifying and solving for x, we get:

x = -3

Therefore, the solution to the system of equations is (x, y) = (-3, 6).

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Therefore , the solution of the given problem of equation comes out to be the value of an is 12 when b is 147. Therefore, 12 is the right response.

In intricate algorithms, variable words are typically employed to demonstrate consistency between two opposing arguments. Equations are academic phrases that are used to demonstrate the equality of different academic figures. Consider the specifics of the above x + 7 suggestions. improving in this case provides b + 7 while constructing y + 7 before integrating.

Here,

The following values can be used to put up a proportion if the relationship between variables a and b is proportional:

=>  a / b = 4 / 49

The value of a when b is 147 can therefore be calculated using this ratio:

=>  a / 147 = 4 / 49

By cross-multiplying, we can isolate a:

=>  a = (4 / 49) * 147

Now that we know the value of a, we can:

=> a = 12

Therefore, the value of an is 12 when b is 147. Therefore, 12 is the right response.

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Answer:

C.0.75

Step-by-step explanation:

the number of occurrence is 15 and the number of simulations is 20 so

P= 15/20

.75

Answer:

12 m

Step-by-step explanation:

Since this is a right triangle, let's use the Pythagorean Theorem.

Recall that the Pythagorean Theorem is:

[tex]a^2+b^2=c^2[/tex]

Where a is the length of one of the legs, b is the length of the other leg, and c is the length of the hypotenuse.

We are given the hypotenuse (15 m) and one of the legs (9 m).

Let's assign some values:

[tex]a=??\\c=15\\b=9[/tex]

Let's solve for b by using the Pythagorean Theorem.

[tex]a^2+b^2=c^2=\\a^2+9^2=15^2=\\a^2+81=225=\\a^2=144=\\a=12[/tex]

So, the length of the missing leg, a, is 12 m.

Answer:

The length of the missing leg is 12 m.

Step-by-step explanation:

SOLUTION :

Here we have given that the base of triangle is 9 m and hypotenuse is 15 m. We have to find the the height of triangle.

Using Pythagoras Theorem to find the height of triangle :

[tex]\quad{\longrightarrow{\sf{{(H)}^{2} = {(b)}^{2} + {(h)}^{2}}}}[/tex]

Substituting all the given values in the formula to find the height of triangle :

[tex]\quad{\longrightarrow{\sf{{(H)}^{2} = {(b)}^{2} + {(h)}^{2}}}}[/tex]

[tex]\quad{\longrightarrow{\sf{{(15)}^{2} = {(9)}^{2} + {(h)}^{2}}}}[/tex]

[tex]\quad{\longrightarrow{\sf{{(15 \times 15)} = {(9 \times 9)} + {(h)}^{2}}}}[/tex]

[tex]\quad{\longrightarrow{\sf{{(225)} = {(81)} + {(h)}^{2}}}}[/tex]

[tex]\quad{\longrightarrow{\sf{{(h)}^{2} = 225 - 81}}}[/tex]

[tex]\quad{\longrightarrow{\sf{{(h)}^{2} = 144}}}[/tex]

[tex]\quad{\longrightarrow{\sf{h = \sqrt{144}}}}[/tex]

[tex]\quad{\longrightarrow{\sf{\underline{\underline{\pink{h = 12 \: m}}}}}}[/tex]

Hence, the height of triangle is 12 m.

The value of the expression is 17/8.

Given is an expression, 7/8 - (-2+3/4) = ( ____+____) + 7/8, we need to solve it,

7/8-(-2+3/4)

= 7/8-(-8+3)/4

= 7/8 + 5/4

= 7+10 /8

= 17/8

Hence, the value of the expression is 17/8.

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There are two buildings that you want to have in the amusement park.

a) The expression that represents the area of the gift shop, in terms of x, is

Area = length x width

Area = (20x + 24) (36x - 20)

Area = 720[tex]x^{2}[/tex] + 208x - 480

Therefore, the area of the gift shop is 720[tex]x^{2}[/tex]  + 208x - 480 square feet.

b) The expression that represents the perimeter of the gift shop, in terms of x, is

Perimeter = 2(length + width)

Perimeter = 2(20x + 24 + 36x - 20)

Perimeter = 2(56x + 4)

Perimeter = 112x + 8

Therefore, the perimeter of the gift shop is 112x + 8 feet.

c) If the perimeter is going to be 176 feet, we can set the expression for the perimeter equal to 176 and solve for x

112x + 8 = 176

Subtracting 8 from both sides

112x = 168

Dividing both sides by 112

x = 1.5

Now that we know x, we can substitute it into the expressions for the length and width of the gift shop

Length = 20x + 24 = 20(1.5) + 24 = 54 feet

Width = 36x - 20 = 36(1.5) - 20 = 34 feet

Therefore, the dimensions of the gift shop are 54 feet by 34 feet.

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Answer:

H = 45 ½ ÷ 6 ½

Step-by-step explanation:

The formula  for the area of the base of the box is L * W.

The formula for the volume of the box is L * W * H

Therefore dividing 45 ½ (volume of box) by 6 ½ (area of base) will give you the height of the box

Craig should make approximately 36 shirts to maximize his earnings for the week.

The maximum value of x for a quadratic equation, ax² + bx + c is given by the formula:

x (max) = -b/2a

Since the weekly income can be modeled by the function p(x)= −2x² + 145x.

In this case a = -2 and b = 145. Thus,
x (max) = -145/2(-2)

           =  145/4

           =  36.25

Thus, Craig should make approximately 36 shirts to maximize his earnings for the week

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The fuel efficiency of the vehicle with the greatest weight which is around 2375kg on this scatter-plot showing vehicle weights & fuel efficiency is 16 mpg.

The relationships between two variables in a data set are represented by graphs or scatter plots.  It displays data points on a Cartesian system or a two-dimensional plane. The dependent variable is represented or plotted on the Y-axis, while the independent variable or attribute is plotted or represented on the X-axis. The scatter plot's objective is to show what happens to one variable when another variable is modified. A theory that the two variables are connected is tested using the scatter plot.

The scatter plot shows the vehicle weights & fuel efficiency.

Data represented on X-axis(independent variable) : weights of vehicle

Data represented on Y-axis(dependent variable) : fuel efficiency.

Lowest weight of vehicle=1125 kg(approx)

Greatest weight of the vehicle=2375 kg(approx)

Lowest fuel efficiencies of surveyed vehicles=16 mpg.

Greatest fuel efficiency of surveyed vehicles=30 mpg.

∴The fuel efficiency of the heaviest vehicle=16 mpg.

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The unit price for a piece of cheese is $0.11 per pound.

To find the unit price, we need to divide the total cost by the total weight. In this case, we have 1.24 pounds of cheese that costs $11.25.

So, we can set up a proportion

1.24 pounds / $11.25 = x pounds / $1

To solve for x, we can cross-multiply

1.24 pounds * $1 = $11.25 * x pounds

1.24 = $11.25 * x

x = 1.24 / $11.25

x = 0.11

So, the unit price is $0.11 per pound.

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The correct statement regarding the best measure of center from the box and whisker plot is given as follows:

The median is the best measure of center and equals 11.

A box and whisker plots shows these five features from a data-set, listed as follows:

The features for the data-set in this problem are given as follows:

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a) Function f(x) = 2^x is an exponential growth function because a = 2 is greater than 0 and b = 2 is greater than 1. The graph of the function starts at (0,1) and increases rapidly as x increases.

b) Function f(x) = (1/2)^x is an exponential decay function because a = 1/2 is greater than 0 and b = 1/2 is between 0 and 1. The graph of the function starts at (0,1) and decreases rapidly as x increases.

Key features of both graphs:

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To find the population of the city, we need to find the total number of people living in the city. We know that there are 1350 people per square mile, so we can find the total number of people by multiplying the number of square miles in the city by the number of people per square mile.

The area of the city is the length times the width, so:

Area = Length x Width

Area = 12 miles x 8 miles

Area = 96 square miles

Now, we can find the population by multiplying the area by the number of people per square mile:

Population = Area x People per square mile

Population = 96 square miles x 1350 people per square mile

Population = 129,600 people

Therefore, the population of the city is 129,600 people.

Hope this helps!