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1. Sasha and Mona practiced swimming every day. They increased the number of minutes of swimming every week as described below:

Sasha: Practiced swimming for 10 minutes every day in the first week, 20 minutes in the second week, 40 minutes in the third week, and 80 minutes in the fourth week.

Mona: Practiced swimming for 10 minutes every day in the first week, 20 minutes in the second week, 30 minutes in the third week, and 40 minutes in the fourth week.

Which statement best describes the methods used by Sasha and Mona to increase the time they practiced swimming?

Mona's method is linear because the number of minutes increased by an equal factor every week.
Sasha's method is exponential because it increases by an equal number every week.
Mona's method is exponential because it increases by a fixed factor, and Sasha's method is linear because it increases by a fixed number.
Sasha's method is exponential because it increases by a fixed factor, and Mona's method is linear because it increases by a fixed number.

2. The sales totals at Macy's food store have increased exponentially over the months. Which of these best shows the sales in the first three months?

$1200 in the first month, $1220 in the second month, $1240 in the third month
$1200 in the first month, $1272 in the second month, $1344 in the third month
$1200 in the first month, $1285 in the second month, $1370 in the third month
$1200 in the first month, $1224 in the second month, and $1248.48 in the third month

Final answer:

To find the month when Pedro's exponentially growing ant farm and Bobby's linearly decreasing ant farm have the same number of ants, we set the exponential growth formula equal to the linear decrease formula and solve for the number of months, using graphical or numerical methods.

Explanation:

We need to determine after how many months Pedro's exponentially growing ant farm will have the same number of ants as Bobby's linearly decreasing ant farm.

Let's denote the number of months as m. Pedro's ant farm starts with 100 ants and grows at a rate of 15% per month, so the population at month m is PPedro = 100 × (1 + 0.15)m.

Bobby's ant farm starts with 350 ants and decreases by 5 ants per month, so the population at month m is PBobby = 350 - 5m.

We need to find the month m where PPedro is approximately equal to PBobby. This can be solved by equating and solving the two expressions:

100 × (1 + 0.15)m ≈ 350 - 5m

To solve this, we can use either graphical methods or numerical methods such as iterative approximation, since an exact solution may not be possible through algebraic methods. We are looking for the value of m that makes both sides approximately equal.

The lowest grade is 48.

The substitution method is the algebraic method to solve simultaneous linear equations. In this method, the value of one variable from one equation is substituted in the other equation.

Let the lowest grade be x and highest grade be y.

According to the given question.

[tex]y = 42 + x[/tex]

Also,

[tex]x + y= 138[/tex]

For finding the points of lowest grade and highest grade substitute

y = 42 + x in x + y = 38.

[tex]x + ( 42 + x ) = 138[/tex]

⇒ [tex]2x + 42 = 138[/tex]

⇒ [tex]x + 21 = 69[/tex]

⇒[tex]x = 69 -21[/tex]

⇒[tex]x = 48[/tex]

Therefore,

[tex]y = 48 + 42 =90[/tex]

Hence, the lowest grade is 48.

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Answer:

  16.9

Step-by-step explanation:

The distance formula can be used to find the lengths of individual segments. It tells you ...

 d = √((Δx)² +(Δy)²)

where Δx and Δy are the differences between x- and y-coordinates of the segment end points.

Since the value is squared, the sign of the difference doesn't matter. It can be easier to write it as always positive, so in some cases it may be Δx = x₂-x₁ and in other cases it might be Δx = x₁-x₂, for example.

If the segments are labeled A, B, C, D, E in order, the distances are ...

 AB = √(5²+1²) = √26 ≈ 5.099

 BC = √(1²+3²) = √10 ≈ 3.162

 CD = Δx = 3

 DE = √(3²+2²) = √13 ≈ 3.606

 EA = Δy = 2

Then the perimeter is ...

 P = AB +BC +CD +DE +EA = 5.099 +3.162 +3 +3.606 +2 = 16.867

 P ≈ 16.9

The average speed in miles per hour for the entire trip of Emily is 38 miles per hour.

The correct answer is Option D.

Given data:

To find Emily's average speed during her entire trip, calculate the total time she spent traveling on both the highway and local roads.

To determine the speed on the highway.

On the highway, she traveled 30 miles faster than on the local roads, and her speed on the local roads is 20 miles per hour.

Speed on the highway = Speed on the local roads + 30 miles per hour

Speed on the highway = 20 miles per hour + 30 miles per hour

Speed on the highway = 50 miles per hour

Now, calculate the time spent on each segment of the trip:

Time spent on the highway = Distance on the highway / Speed on the highway

Time spent on the highway = 60 miles / 50 miles per hour

Time spent on the highway = 1.2 hours

Time spent on local roads = Distance on local roads / Speed on local roads

Time spent on local roads = 16 miles / 20 miles per hour

Time spent on local roads = 0.8 hours

Now, add the times for both segments to get the total time:

Total time = Time spent on the highway + Time spent on local roads

Total time = 1.2 hours + 0.8 hours

Total time = 2 hours

So, the average speed is determined as:

Average speed = Total distance / Total time

Average speed = (60 miles + 16 miles) / 2 hours

Average speed = 76 miles / 2 hours

Average speed = 38 miles per hour

Hence, Emily's average speed during her entire trip was 38 miles per hour.

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Answer:

h(x), f(x) , g(x)

Step-by-step explanation:

[tex]f(x) = 2x^2 - 8x[/tex]

Axis of symmetry at [tex]x=\frac{-b}{2a}[/tex], a=2, b=-8

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

[tex]x=\frac{-(-8)}{2(2)}=2[/tex]

[tex]g(x) = x^2 - 6x + 1[/tex], a= 1, b=-6

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

[tex]x=\frac{-(-6)}{2(1)}=3[/tex]

[tex]h(x) = -2x^2[/tex], a=-2, b=0

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

[tex]x=\frac{-(0)}{2(-2)}=0[/tex]

Now rank the function based on their axis of symmetry

h(x), f(x) , g(x)

The angle which is 24° more than the measure of its complementary angle is 66°.

Two angles whose sum is equivalent to 90 degrees are called complementary angles. Mathematically -

∠x + ∠y = 90°

Assume that the angle which is 24° more than the measure of its complementary angle be [x].

Then, we can write -

x + 24 = 90

x = 90 - 24

x = 66°

Therefore, the angle which is 24° more than the measure of its complementary angle is 66°.

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Two step algebraic equations are relatively quick and easy -- after all, they should only take two steps. To solve a two step algebraic equation, all you have to do is isolate the variable by using either addition, subtraction, multiplication, or division. If you want to know how to solve two step algebraic equations in a variety of ways, just follow these steps.

1. Write the problem. The first step to solving a two step algebraic equation is just to write the problem so you can start to visualize the solution. Let's say we're working with the following problem: -4x + 7 = 15.

2. Decide whether to use addition or subtraction to isolate the variable term. The next step is to find a way to keep "-4x" on one side and to keep the constants (whole numbers) on the other side. To do this, you'll have to do the "Additive Inverse," finding the opposite of +7, which is -7. Subtract 7 from both sides of the equation so that the "+7" on the same side as the variable term is canceled out. Just write "-7" below the 7 on one side and below the 15 on the other so the equation remains balanced. Remember the Golden Rule of Algebra. Whatever you do to one side of an equation must be done to the other side to maintain the balance. That is why 7 is subtracted from the 15 as well. We only need to subtract 7 once per side, which is why the 7 is not subtracted from the -4x as well.

To solve two-variable equations, identify unknowns, find related equations, substitute known values, and solve for one variable before finding the other. Use substitution or elimination methods in case of a system of equations.

To solve equations with 2 variables, follow these steps:

Regarding a 2x2 system (two equations with two unknowns), you can either use substitution—solving one equation for one variable and then substituting into the other—or elimination—adding or subtracting equations to cancel one of the variables. If an equation has more than one unknown, additional equations are necessary to solve the system.

For instance, consider a pair of equations:
1) x + y = 10
2) x - y = 6

By adding these two equations, y is eliminated, and we can solve for x quickly. Then, we can substitute the value of x into either equation to solve for y.

Final answer:

The rule that represents the given function with ordered pairs (1, 4), (2, 16), (3, 64), (4, 256), (5, 1,024) is f(x) = 4^(x-1), which is determined by observing the pattern of the ordered pairs.

Explanation:

Identifying the Rule of a Function

The ordered pairs (1, 4), (2, 16), (3, 64), (4, 256), and (5, 1,024) represent a function where each input from the domain is mapped to a unique output in the range. Observing the pattern of the ordered pairs, we can infer a rule that corresponds to this function. Each second element in the pair appears to be an exponentiation of 4 to the power of the first element minus one.

The general rule for this function can be expressed as f(x) = 4^(x-1). We can verify this by plugging in the values from the domain (1 through 5) into the function:

For x = 1, f(1) = 4^(1-1) = 4^0 = 1

For x = 2, f(2) = 4^(2-1) = 4^1 = 4

For x = 3, f(3) = 4^(3-1) = 4^2 = 16

For x = 4, f(4) = 4^(4-1) = 4^3 = 64

For x = 5, f(5) = 4^(5-1) = 4^4 = 256

Therefore, the rule representing this function is f(x) = 4^(x-1).

Final answer:

[tex]\[ h(2) = 55, \quad h(-1) = 3 \][/tex]

Explanation:

To evaluate [tex]\(h(x)\)[/tex] at the given values using synthetic division, we start with the coefficients of the polynomial [tex]\(x^6 - 4x^4 + 5x^2 + 1\).[/tex]

1. For [tex]\(h(2)\):[/tex]

  - Write down the coefficients: 1, 0, -4, 0, 5, 0, 1.

  - Using synthetic division with 2 as the divisor, perform the operations to obtain the result 55.

2. For [tex]\(h(-1)\):[/tex]

  - Repeat the process with -1 as the divisor, using the coefficients: 1, 0, -4, 0, 5, 0, 1.

  - After the calculations, the result is 3.

In both cases, the final values represent the result of evaluating the function [tex]\(h(x)\)[/tex] at x = 2 and [tex]\(x = -1\)[/tex] respectively.

By setting up the proportion [tex]\(\frac{15}{x} = \frac{x}{135}\)[/tex] and solving for [tex]\( x \)[/tex], we find that [tex]\( x = 45 \).[/tex]

To find the value of [tex]\( x \)[/tex] when the terms 15, [tex]\( x \)[/tex], and 135 are in proportion, we set up the proportion:

[tex]\[\frac{15}{x} = \frac{x}{135}\][/tex]

This represents the relationship where the first ratio is equal to the second ratio. Cross-multiplying, we get:

[tex]\[15 \times 135 = x \times x\][/tex]

[tex]\[2025 = x^2\][/tex]

Taking the square root of both sides, we find:

[tex]\[x = \sqrt{2025}\][/tex]

[tex]\[x = \pm 45\][/tex]

However, since 15,  x , and 135 are in proportion, the value of  x  must be positive. Therefore,  x = 45 .

The question probable maybe:

If the three terms 15, x and135 are in proportion then what is the value of x