Multiply (2.4 ⋅ 1014) ⋅ (4 ⋅ 107). Express the answer in scientific notation. 9.6 ⋅ 1021 9.6 ⋅ 1022 96 ⋅ 1021 96 ⋅ 1022

Answer:  The required number of servings in 6 cups of yogurt is 9.

Step-by-step explanation:  We are given to find the number of [tex]\dfrac{2}{3}[/tex] cup servings in 6 cups of yogurt.

We will be using the UNITARY method to solve the given problem.

We have

Number of servings in [tex]\dfrac{2}{3}[/tex] cups of yogurt = 1.

So, the number of servings in 1 cup of yogurt will be

[tex]\dfrac{1}{\frac{2}{3}}=\dfrac{3}{2}.[/tex]

Therefore,  the number of servings in 6 cups of yogurt is given by

[tex]\dfrac{3}{2}\times6=9.[/tex]

Thus, the required number of servings in 6 cups of yogurt is 9.

Answer:

Option B is the correct option.

Step-by-step explanation:

The Linear function is always in the form of y = mx + c

Where m defines the slope which is rate of change of y-values or change in y-values with the change in x-values.

Therefore, Option B is the correct option.

Final answer:

To divide 55 by 2044, use long division. The quotient is 0.027.

Explanation:

To divide 55 by 2044, you can use long division. Here is the step-by-step process:

At this point, you have reached the end of the division process. The final quotient is 0.027 and there is no remainder. Therefore, 55 divided by 2044 is approximately 0.027.

Final answer:

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. When multiplying or dividing with negative exponents, add or subtract them accordingly. In scientific notation, divide the coefficients and subtract the exponents to divide numbers.

Explanation:

Working with negative exponents is an essential skill in mathematics, where a negative exponent indicates the inverse or reciprocal of the base raised to the corresponding positive exponent. For example, 2-1 equals 1/2, which means that we are finding the reciprocal of 2. In operations involving exponents, such as multiplication and division, we follow certain rules with negative exponents:

When multiplying numbers, add the exponents (negative exponents included).

When dividing numbers, subtract the exponents (again, include negative exponents).

To convert a negative exponent into a positive one, take the reciprocal of the base with the corresponding positive exponent.

Here's an example of division in scientific notation: 106 ÷ 103 = 10(6-3) = 103. If you need to divide two numbers using scientific notation, you divide the coefficients and subtract the exponents.

Remember, a number like 3-4 is equivalent to 1/34, or 1 divided by 3 multiplied by itself four times. In general, negative exponents flip the base to the denominator, signifying a division rather than multiplication.

The solution of the equation 72x = 1 is 1/72 and 70x = 1 is 1/70.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

Making anything easier to accomplish or comprehend, as well as making it less difficult, is the definition of simplification.

The equation is given below.

72x = 1

Simplify the equation, then the value of x will be

72x = 1

x = 1/72

The equation is given below.

70x = 1

Simplify the equation, then the value of x will be

70x = 1

x = 1/70

Then the solution of the equation 72x = 1 is 1/72 and 70x = 1 is 1/70.

More about the Algebra link is given below.

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Answer:

A.[tex]PR^2=PQ^2+QR^2[/tex]

Because [tex]25=9+16[/tex]

Step-by-step explanation:

We are given that

Area of square A=[tex]9[/tex] square units

Are of square B=[tex]16[/tex] square units

Area of square C=[tex]25[/tex] square units

We have to find the relationship between sides of triangle PQR.

Side of square=[tex]\sqrt{Area\;of\;square}[/tex]

Using the formula

Side of square A=[tex]\sqrt{9}[/tex]=3 units

Side of square B=[tex]\sqrt{16}=4[/tex] units

Side of square C=[tex]\sqrt{25}=5[/tex] units

Pythagoras theorem

[tex](hypotenuse)^2=(Base)^2+(Perpendicular\;side)^2[/tex]

Using Pythagoras theorem

[tex]PR^2=PQ^2+QR^2[/tex]

[tex]5^2=3^2+4^2[/tex]

[tex]25=9+16[/tex]

Option A is true

The common ratio between each pair of terms in the given geometric sequence of fractions is -2/5.

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant factor called the common ratio. To find the common ratio in a geometric sequence of fractions, we divide each term by the previous term. Let's calculate the common ratios for the given sequence:

As we can see, the common ratio between each pair of terms in the given geometric sequence of fractions is -2/5.

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The two equations are:

g= w +8

g= w +2

^_^

At the end of 1 year, the amount in the account will be $9860. At the end of 2 years, the amount in the account will be $11437.6. This is calculated using the formula for compound interest.

Lisa is placing $8500 in an account that pays 16% interest compounded annually. To solve for part (a) and part (b) which ask for the amount in the account at the end of 1 year and 2 years respectively, we use the formula for compound interest: A = P *(1 + r/n)^(nt).

Here, P is the principal amount ($8500), r is the annual interest rate (16% or 0.16 as decimal), n is the number of times that interest is compounded per unit t (n = 1 as it's compounded annually), and t is the time the money is invested for.

For (a), t = 1 year. Inserting these values into the formula, we get:

A = $8500 * (1 + 0.16/1)^(1*1) = $8500 * 1.16 = $9860.

Therefore, at the end of 1 year, the amount in the account will be $9860.

For (b), t = 2 years. Using the same formula, we get:

A = $8500 * (1 + 0.16/1)^(1*2) = $8500 * 1.16^2 = $11437.6.

Therefore, at the end of 2 years, the amount in the account will be $11437.6.

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Line x =2 has undefined slope.

" Slope is defined as the ratio of the difference in the increase of  y-coordinates to the difference in the increase of  x-coordinates."

Formula used

Equation of the line

y = mx + c

c = y-intercept

m = slope

m = [tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

According to the question,

1. y = 0

Compare it with standard equation of the line

y = 0x + 0

slope = 0

It is not a correct answer.

2. y =2

Compare it with standard equation of the line

y = 0x + 2

slope = 0

It is not a correct answer.

3. x= y

Compare it with standard equation of the line

y = 1x + 0

slope = 1

It is not a correct answer.

4. x=2

It is vertical line , x value remain constant.

Therefore,

[tex]x_{1}=2\\x_{2}=2[/tex]

Slope = [tex]\frac{y_{2}-y_{1}}{2-2}[/tex]

         = [tex]\frac{y_{2}-y_{1}}{0}[/tex]

Zero in the denominator therefore, undefined slope

Hence, line x =2 has undefined slope.

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Final answer:

The line with an undefined slope is D. x = 2, which is a vertical line with a constant x-value for all points on the line.

Explanation:

The question asks which line has an undefined slope. An undefined slope occurs when a line is vertical since slope is defined as the change in y over the change in x, and in a vertical line the change in x is zero. Looking at the options given, the equation representing a vertical line is D. x = 2. This line is vertical because it has the same x-value for all points on the line, meaning there is no change in x as you move along the line, thus creating an undefined slope.

Let's break this down. We know that there are 5 weight machines, and Leo spent an equal amount of time on all of them. Since we don't know how long he spent on each, we'll use x. Our equation so far is 5x. Then, we know that he spent 2/3 hour on the treadmill, which we will just add to the 5x. So, now our equation is 5x + 2/3. Lastly, we know that the total time Leo spent at the gym is 1 3/4 hours, which we set the last equation equal to. Our final equation is 5x + 2/3 = 1 3/4. Then, all we have to do is solve for x. What we can do to make this easier is to change all of the fractions into numbers. After changing everything, here is our equation: 5x + 40 = 105. Then, we will subtract 40 from each side and divide both sides by 5. Therefore, Leo spent 13 minutes on each weight machine.

ANSWER:

A) 5x + 2/3 = 1 3/4 ; Leo spent 13 minutes at each machine

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Final answer:

Solving for the time Leo spent on each weight machine involves setting up an equation based on the total gym time and subtracting the time spent on the treadmill, resulting in Leo spending about 13 minutes at each weight machine.

Explanation:

To solve for the amount of time Leo spent at each weight machine, we need to set up an equation based on the total time spent at the gym. First, we know that Leo spent 2/3 of an hour on the treadmill. With a total gym time of 1 3/4 hours (which is the same as 7/4 hours), we can subtract the treadmill time to find the time spent on the weight machines. Thus, the equation can be set up as follows:

5x + 2/3 = 7/4

To solve for x, the hours Leo spent at each weight machine, follow these steps:

Subtract 2/3 from both sides of the equation: 5x = 7/4 - 2/3

Find a common denominator for the fractions and subtract: 5x = 21/12 - 8/12

Simplify the right side: 5x = 13/12

Divide both sides by 5 to solve for x: x = 13/60 hours

This means Leo spent 13/60 hours, or about 13 minutes, at each of the 5 weight machines.

Answer:

The required inequality that shows the maximum number of packages is [tex]n\leq 67[/tex].

Step-by-step explanation:

Consider the provided information.

Each package weighs 3 pounds and Jacob weighs 99 pounds.

let n is the maximum number of package that Jacob can carry with himself in the elevator.

Thus, the total weight of Jacob while having n packages on the elevator is:

[tex]3n+99[/tex]

The elevator can carry a maximum of 300 pounds at a time, therefore.

[tex]3n+99\leq 300[/tex]

Now, subtract 99 from both sides and divide the obtained inequality with 3.

[tex]3n+99-99\leq 300-99[/tex]

[tex]3n\leq 201[/tex]

[tex]\frac{3n}{3}\leq \frac{201}{3}[/tex]

[tex]n\leq 67[/tex]

Thus, the Jacob can carry maximum 67 packages.

Hence, the required inequality that shows the maximum number of packages is [tex]n\leq 67[/tex].

To maintain the scale of a children's soccer net to that of a regulation net, the height should be 2 feet, given that the child-size net is 6 feet wide. The scale factor used is 0.25.

The question asks us to find the scaled dimensions for a children's soccer net based on the dimensions of a regulation soccer goal net. The original dimensions are 24 ft in width and 8 ft in height. If a children's soccer net is 6 ft wide and we want to keep the same scale, we need to calculate the height of the net in proportion to the regulation net's dimensions.

Height Calculation:

The scale factor is the ratio of the width of the children's net to the width of the regulation net. To find the height of the children's net, we multiply the height of the regulation net by the scale factor:

Scale Factor = Width of Children's Net / Width of Regulation Net

Scale Factor = 6 ft / 24 ft = 0.25

Height of Children's Net = Scale Factor × Height of Regulation Net

Height of Children's Net = 0.25 × 8 ft = 2 ft

Therefore, to maintain the scale, the height of the children's soccer net should be 2 feet, and the scale factor is 0.25.