A car rental cost $80 per day, for the first 250 miles, plus $0.20 for each additional mile. If Sonya's rental cost was $124 for one-day rental, how many miles did she drive?

The​ half-life of this radioactive isotope is 135.897 days and this can be determined by using the formula of half-life.

Given :

The formula of the half-life is given by:

[tex]\rm A =P \left(\dfrac{1}{2} \right)^{\dfrac{t}{t_{1/2}}}[/tex]

where A is the final concentration, P is the initial concentration, t is the time period, and [tex]\rm t_{1/2}[/tex] is the half-life.

Now, substitute the known values in the above formula.

[tex]\rm 0.13P =P \left(\dfrac{1}{2} \right)^{\dfrac{400}{t_{1/2}}}[/tex]

Simplify the above expression.

[tex]\rm 0.13 = \left(\dfrac{1}{2} \right)^{\dfrac{400}{t_{1/2}}}[/tex]

Take the log on both sides.

[tex]\rm log(0.13) = {\dfrac{400}{t_{1/2}}} \times log\left(\dfrac{1}{2} \right)[/tex]

[tex]\rm t_{1/2} = \dfrac{400\times log(0.5)}{log(0.13)}[/tex]

Simplify the above expression in order to evaluate the [tex]\rm t_{1/2}[/tex].

[tex]\rm t_{1/2} = 135.897\;days[/tex]

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The half-life of a radioactive isotope is calculated by using the decay formula and solving for the known half-life, given that 13% of the original isotope remained after 400 days.

To determine the half-life of a radioactive isotope, we first need to understand that half-life is the time required for half of a sample of a radioactive isotope to decay. From the problem, it stated that after 400 days, only 13% of the original isotope remained. The formula for decay is N = N0 * (1/2)^(t/T), where N is the final amount, N0 is the initial amount, t is the time elapsed, and T is the half-life.

Given that N/N0 is 0.13 after 400 days, we can rewrite the decay formula to solve for T. By substituting the values into the equation, i.e., 0.13 = (1/2)^(400/T), we can use the laws of logarithms to solve for T. So then we will get the half-life (T) of the isotope.

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The new exponent of the product is -7

From the question, we have the following parameters that can be used in our computation:

6.5 x 10^-8 is multiplied by 10.

This means that

Product = 6.5 x 10^-8 *  10

Evaluate the product

So, we have

Product = 6.5 x 10^-7

From the above, we have

Exponent = -7

Hence, the new exponent is -7

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The mathematical  sentence written with a greater than, a less than sign, or a not equal sign is know as Inequality.

Inequalities are mathematical formulas in which neither side is equal. Unlike equations, we compare two values in inequality. In between, the equal sign is substituted by a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

As, the mathematical  sentence written with a greater than, a less than sign, or a not equal sign.

The definition of inequality is that two things are NOT equal. One of the items could be less than, greater than, less than or equal to the other things, or greater than or equal to the other things.

So, the statement is Inequality statement.

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To find the individual cost of a hot dog and hamburger from a concession stand, we set up a system of equations from the given information and solve it to get the cost of one hot dog as $1.25 and one hamburger as $2.00.

To find the cost of one hot dog and one hamburger, we set up a system of equations based on the information given:

Let's denote the cost of one hot dog as d and the cost of one hamburger as h. We can then rewrite the given information as:

Solving this system of equations, we can multiply the first equation by 2 and the second equation by 3, and subtract them, cancelling out the hamburger terms:

Subtracting the first equation from the second gives:

Thus, one hamburger costs $2.00. Plugging this back into the first equation, we get:

Which simplifies to:

Therefore, one hot dog costs $1.25.

The rate of change of y with respect to x is 35 words per minute.

The rate of change of y with respect to x can be found by taking the slope of the line that represents the relationship between x and y. To find the slope, we can use the formula:

slope = change in y / change in x

Using the given information, we can calculate the slope as:

slope = (525 - 140) / (15 - 4) = 385 / 11 = 35

Therefore, the rate of change of y with respect to x is 35 words per minute.

Final answer:

Jeremy's rate of change in typing speed is 35 words per minute, calculated by dividing the change in words typed by the change in time (385 words/11 minutes).

Explanation:

The student is asking about the rate of change of the words typed with respect to time spent typing. To calculate this, we will use the given values: 140 words in 4 minutes and 525 words in 15 minutes. The rate of change, essentially being the slope of the line that represents the relationship between words typed (y) and the time (x), can be calculated by finding the ratio of the change in words to the change in time:

This means that Jeremy types at an average speed of 35 words per minute.

To find the different options for a car model with multiple customizable features, you multiply the number of choices for each feature. For this model, with 10 colors, 2 options each for transmission, four-wheel drive, and air conditioning, and 3 speaker options, there are 240 different combinations.

To calculate the number of different options for this model of a car, we need to consider all the features that can be selected and multiply the number of choices for each feature together. For the color, there are 10 options. For each of the features, such as automatic transmission, four-wheel drive, and air conditioning, there are 2 choices: with or without. Finally, for the radio-CD speakers, there are 3 options: two, three, or four speakers. Multiplying these choices together gives us the total number of different options:

10 (colors) × 2 (transmission) × 2 (four-wheel drive) × 2 (air conditioning) × 3 (radio-CD speakers) = 240 different options.

So, there are 240 different options available for this model of a car.

Final answer:

To find the total number of different options for the car, we multiply the number of choices for each feature resulting in 240 different options.

Explanation:

The student is asking about the number of different option combinations for a specific model of a new car. The car can be customized with ten colors, automatic transmission or not, four-wheel drive or not, air conditioning or not, and can have two, three, or four radio-CD speakers. To determine the total number of different options, we multiply the number of choices for each feature:

Therefore, the total number of different options is calculated as follows:

10 (colors) × 2 (transmission) × 2 (drive) × 2 (air conditioning) × 3 (speakers)

= 240 different options.

It will take her 13 weeks to save enough money for a trip to the Six Flags amusement park

Comparison is an effort to compare two or more objects in terms of shape or size, or number  

Proportional Comparisons / Directly proportional are comparisons of two or more numbers where one number increases, the other numbers also increase at the same rate  

 Can be formulated  

[tex]\displaystyle \frac {x1} {y1} = \frac {x2} {y2}[/tex]

 so that if:  

 x = 2  

 then  

[tex]\displaystyle y ={ \frac {y} {x} \times \: 2}[/tex]

 While the reversal value comparison / inversely proportional is the comparison of two or more numbers where one number increases, the other number decreases in value  

or when one value decreases at the same rate that the other increases  

Can be formulated  

[tex]\displaystyle \frac {x1} {y2} = \frac {x2} {y1}[/tex]

so that if:  

 x = 2  

 then  

[tex]\displaystyle y = \frac {x} {y} \times \: 2[/tex]

Michelle wants to save $150

she saves $12 each week

the number of weeks needed to save money

every weeks , she save $12 or we can make equation 1:

1 weeks = $12

the amount of money needed to travel to the six flags amusement park

= $150

we can make equation 2

? weeks = $150

Because this includes Proportional Comparisons / Directly proportional,

from two equation above , we can make comparison :

[tex]\displaystyle \frac{1\:weeks}{\$12} =\frac{x\:weeks}{\$150}[/tex]

[tex]\displaystyle x\:weeks=\frac{\$150\times1\:weeks}{\$12}[/tex]

[tex]\large{\boxed{x\:weeks= 12.5}}[/tex]

or we round up to :

[tex]\large{\boxed{\bold{13\:weeks}}}[/tex]

1,092 flowers are arranged into 26 vases  

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dividing 878 by 31  

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round trips  

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The calories in a 21-oz iced tea that is originally 78 calories in a 13-oz serving, multiply the calories per ounce by the new serving size for a total of 126 calories.

The 13-oz iced tea contains 78 calories. To find the calories in a 21-oz iced tea:

Calculate the calories per ounce: 78 calories ÷ 13 oz = 6 calories/oz

Multiply the calories per ounce by the 21-oz serving size: 6 calories/oz × 21 oz = 126 calories in a 21-oz iced tea

we have that

[tex] f(x) =x^{2} + x - 9 [/tex]

Using a graph tool

see the attached figure

The function represent a vertical parabola that open up

the vertex is a minimum-------> is the point [tex] (-0.5,-9.3) [/tex]

The x-intercepts are the points when the y coordinate is equal to zero

The x-intercepts are the points [tex] (-3.5,0) [/tex] and [tex] (2.5,0) [/tex]

so

the function cross the negative x-axis at point [tex] (-3.5,0) [/tex]

therefore

the answer is

(–4, 0) and (–3, 0)

The point that is closest to the point (3, 0) can be found by finding the

minimum value for the function of the distance between points.

The closest point on the hyperbola  x·y = 8 to the point (3, 0) is the point [tex]\underline{(4, \ 2)}[/tex]

Reasons:

The equation of the hyperbola is x·y = 8

The given point = (3, 0)

Therefore;

[tex]y = \dfrac{8}{x}[/tex]

The distance of a point from a line, [tex]d = \mathbf{ \sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}}[/tex]

Where;

(x₁, y₁) = (3, 0)

(x₂, y₂) = The closest point on the hyperbola

We get;

[tex]d = \sqrt{\left (y-0 \right )^{2}+\left (x-3 \right )^{2}}[/tex]

Which gives;

[tex]d = \sqrt{\left (\dfrac{8}{x} -0 \right )^{2}+\left (x-3 \right )^{2}} = \mathbf{ \sqrt{\left (\dfrac{8}{x}\right )^{2}+\left (x-3 \right )^{2}}}[/tex]

At the minimum distance, using an online app, we have;

[tex]\dfrac{d}{dx}d = \dfrac{d}{dx} \left( \sqrt{\left (\dfrac{8}{x}\right )^{2}+\left (x-3 \right )^{2}} \right) = \dfrac{-64 + x^3 \cdot (x - 3)}{x^2 \cdot \sqrt{x^2 \cdot (x - 3)^2 + 64} } = 0[/tex]

Which gives;

x³·(x - 3) - 64 = 0

Using a graphing calculator, we get;

(x - 4)·(x³ + x² + 4·x + 16) = 0

Therefore;

A solution for the closest or point of minimum distance is x = 4

Which gives;

[tex]y = \dfrac{8}{4} = 2[/tex]

Therefore, the closest point on the hyperbola  x·y = 8 to the point (3, 0) is the

point [tex]\underline{(4, \ 2)}[/tex]

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The length of the rectangle with a perimeter of 36 meters and a length that is 2 meters more than three times the width is 14 meters.

The perimeter of a rectangle is twice the sum of its length and width. According to the question, the perimeter of the rectangle is 36 meters, and the length (L) is 2 meters more than three times the width (W). We can express this relationship as follows: L = 3W + 2.

The formula for the perimeter (P) of a rectangle is P = 2L + 2W.

Using the given perimeter of 36 meters, we can set up the equation as 2(3W + 2) + 2W = 36.

Simplifying this equation, we find that 8W + 4 = 36, and thus W = 4 meters.

Substituting the value of W back into the equation for L gives us L = 3(4) + 2, which means the length of the rectangle is 14 meters.

To find the length of the ladder, we can use the property of right triangles. The length of the ladder is 10 feet.

To find the length of the ladder, we can use the property of right triangles. Let's assume the length of the ladder is 'L'.

According to the given information, the distance from the ground to the top of the ladder is 2 feet less than the length of the ladder. So, the height of the triangle formed by the ladder, the building, and the ground is L - 2.

The distance from the bottom of the ladder to the building is given as 6 feet. Using the Pythagorean theorem, we can set up the following equation: (L-2)^2 + 6^2 = L^2

Simplifying the equation, we get L^2 - 4L + 4 + 36 = L^2, which simplifies further to 40 = 4L. Dividing both sides by 4, we get L = 10.

Therefore, the length of the ladder is 10 feet.

Final answer:

The ratio of red cards to other cards that Brent counted (10 red cards, 10 black cards, 20 blue cards) is 1:3 after summing black and blue cards to find the total number of other cards and simplifying the ratio.

Explanation:

Brent counted 10 red cards, 10 black cards and 20 blue cards. The question is what is the ratio of red cards to other cards. To calculate the ratio, we need to determine the total number of cards that are not red, which would be the black and blue cards combined.

First, we sum the black and blue cards:

10 black cards + 20 blue cards = 30 other cards.

Now, we set up the ratio of red cards to other cards:

10 red cards : 30 other cards, which simplifies to 1 : 3 when both sides of the ratio are divided by 10, the greatest common divisor.

Therefore, the ratio of red cards to other cards is 1:3.

Answer:

There are 14 bags in 3.5 boxes.

Step-by-step explanation:

Given : Lettuce is packaged in four bags to a box. If there are 3.5 boxes of lettuce in a cooler.

To find : How many bags of lettuce would there be?

Solution :

Lettuce is packaged in four bags to a box.

i.e, In 1 box there are 4 bags.

If there are 3.5 boxes of lettuce in a cooler.

In 3.5 boxes number of bags are [tex]4\times 3.5=14[/tex]

Therefore, There are 14 bags in 3.5 boxes.

Answer:

5 people

Step-by-step explanation:

Let

The total cost was $1400, which was to be divided equally among the members going. Then,

y = 1400 / x [1]

Prior to the trip, 3 members decided not to go. The cost per person increased by $105.

y + 105 = 1400 / (x - 3)

(y + 105) . (x - 3) = 1400 [2]

We replace [1] in [2],

(1400/x + 105) . (x - 3) = 1400

1400 - 4200/x + 105x -315 = 1400

- 4200/x + 105x = 315

(- 4200 + 105x²)/x = 315

- 4200 + 105x² = 315 x

105x² - 315x - 4200 = 0

When we solve this quadratic equation using the quadratic formula, we get x₁ = 8 and x₂ = -5. Only the positive x has sense.

The people that went to the expedition was x - 3 = 8 - 3 = 5.