An item priced at $200 has a fee of $12. An item priced at $150 has a fee of $9. The restocking fee for an item priced at $180 is $10.
To solve this problem, we need to use a proportion to relate the item price to the restocking fee. The problem provides two pairs of corresponding item prices and restocking fees: $200 and $12, and $150 and $9. We can set up a proportion using these pairs:
$200/$12 = $150/$9
We can simplify this proportion by dividing both sides by 3:
$200/$4 = $150/$3
Now we can use the simplified proportion to find the restocking fee for an item priced at $180. We'll use cross-multiplication:
$200 x $3 = $150 x $4
$600 = $600
This tells us that the ratios are equivalent, and the restocking fee for an item priced at $180 is $10.
In other words, the restocking fee is proportional to the item price, and we can use the given pairs of prices and fees to find the fee for any given price using a proportion.
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1 Aiden paints an unassembled gift box to use for
his sister's birthday gift.
The base of the box measures 2 inches by5 inches. When assembled the box measures7 inches tall. If Aiden only paints the outside of the unassembled box, how many square inches will he paint?
A 64 in.²
B 70 in.²
C 118 in.²
D 126 in.²
A water tower is 16 meters tall. Close by, there is a crane that is 24 meters tall. If the water tower's shadow is 30 meters long, how long is the crane's shadow?
The crane's shadow is 45 meters long when the crane is 24 meters tall.
What are similar triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle resemblance is indicated here by the symbol "≅"
Let the shadow cast by the crane be x.
The triangle formed by the tower and by the crane are similar.
Using the rule of similar triangles we have:
16/24 = 30/x
16x = (30)(24)
x = 45
Hence, the crane's shadow is 45 meters long.
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an urn contains 3 red marbles, 4 green marbles, and 5 black marbles. three marbles are drawn at random. find the probability of getting exactly one red marble or exactly one green marble and not both at the same time.
The answer is that there is no probability for this event to happen. To find the probability of getting exactly one red marble or exactly one green marble and not both at the same time, we can use the following steps:
Step 1: Find the number of ways to draw 3 marbles out of 12.
This can be found using the combination formula, which gives:
C(12,3) = 12! / (3! * 9!) = 220
Therefore, there are 220 ways to draw 3 marbles out of 12.
Step 2: Find the number of ways to draw exactly one red marble and two non-red marbles.
There are 3 red marbles, and we need to choose one, so there are C(3,1) = 3 ways to do this. There are 9 non-red marbles, and we need to choose two of them, so there are C(9,2) = 36 ways to do this. Therefore, 3 * 36 = 108 ways to draw exactly one red marble and two non-red marbles.
Step 3: Find the number of ways to draw exactly one green marble and two non-green marbles.
There are 4 green marbles, and we need to choose one, so there are C(4,1) = 4 ways to do this. There are 8 non-green marbles, and we need to choose two of them, so there are C(8,2) = 28 ways to do this. Therefore, there are 4 * 28 = 112 ways to draw exactly one green marble and two non-green marbles.
Step 4: Find the number of ways to draw exactly one red marble and one green marble. There are 3 red marbles and 4 green marbles, and we need to choose one of each, so there are 3 * 4 = 12 ways to do this.
Step 5: Add the results from steps 2, 3, and 4 to get the number of ways to draw marbles that satisfy the given condition.
Total number of ways = 108 + 112 + 12 = 232
Step 6: Find the probability of getting exactly one red marble or one green marble but not both at the same time.
The probability of an event is the number of ways that event can happen divided by the total number of possible outcomes. Therefore, the probability of getting exactly one red marble or exactly one green marble and not both at the same time is:
P = (number of ways to draw marbles that satisfy the given condition) / (total number of ways to draw 3 marbles out of 12 marbles)
P = 232 / 220
P = 1.0545
However, probabilities cannot be greater than 1. Therefore, the answer is that there is no probability for this event to happen.
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State how the triangles are congruent using SSS, SAS, ASA, AAS, or
HL. If they are not congruent, type NOT.
Answer:
AAS
Step-by-step explanation:
[tex]\overline{PR} \cong \overline{PR}[/tex] by the reflexive property.
What expression is equivalent to 24a+(-26b)-13a+12b?
The expression 24a+(-26b)-13a+12b is equivalent to 11a-14b.
What is algebraic expression ?
An algebraic expression is a combination of variables, numbers, and mathematical operations, such as addition, subtraction, multiplication, and division. It can be used to represent a mathematical relationship or formula and can be simplified or evaluated using algebraic rules.
Examples of algebraic expressions include "3x + 4y", "2a^2 - 5b", and "(x + 3)(x - 2)".
Given expression ,
24a+(-26b)-13a+12b
= 24a - 13a +12 b - 26b
= 11a - 14b
Therefore, The expression 24a+(-26b)-13a+12b is equivalent to 11a-14b.
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10.ja box contains 24 light bulbs, of which two are de- feefive. if a person selects 10 bulbs at random, without replacement, what is the probability that both defective bulbs will be selected?
Hence, the probability that both defective bulbs will be selected is 0.1630
We are given that a box containing 24 light bulbs. 2 light bulbs are defective.
Now if a person selects 10 bulbs at random, without replacement.
We are required to find the probability that both defective bulbs will be selected.
Since there are two defectives, therefore two defective can be chosen in(2,2) ways.
Also since 10 bulbs are selected, the remaining 8 bulbs can be selected in (22,8)
Also, the total number of ways 10 bulbs can be selected is:
[tex]^{22}C_8\\[/tex]
Therefore, the probability that both defective bulbs will be selected is:
[tex]=\frac{(^2c_2)(^{22}C_2)}{^{24}C_10}\\\\=1.630.[/tex]
rounded to 4 decimal places
Hence, the probability that both defective bulbs will be selected is 0.1630
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How to write a congruence statement
Answer:
Step-by-step explanation: A congruence statement expresses that two figures have the same size and shape. Here's how you can write a congruence statement step by step:
Write the two figures you want to compare. For example, you could write "Triangle ABC" and "Triangle DEF".
Use the symbol "≅" to indicate congruence. Place the symbol between the two figures, like this: "Triangle ABC ≅ Triangle DEF".
Label the corresponding parts of the two figures to show that they are congruent. For example, you could write "AB = DE", "BC = EF", and "AC = DF".
The final congruence statement would look like this: "Triangle ABC ≅ Triangle DEF, with AB = DE, BC = EF, and AC = DF". This statement says that Triangle ABC is congruent to Triangle DEF, and that the lengths of their corresponding sides are equal.
what is the rate constant at 42.4 ∘c based on the data collected for trial e?
The rate constant at 42.4 based on trial E is Value[k = 2.68 X 10^-3] and Units[M-1 s-1]
The rate constant at 42.4 0C is [acetone] initial = 0.2 M, [H+] initial = 0.01 M, so rate= k [acetone] [H+] 5.37 X 10^-6 = k X 0.2 X 0.01, k = 2.68 X 10^-3 , and the order of the reaction is two therefore the unit of the speed constant is M-1 s-1. reaction rate constants and rate laws is a fundamental concept in chemical kinetics, which is the study of the rates and mechanisms of chemical reactions. Reaction rate constants are specific to a particular reaction and are influenced by several factors, including temperature, concentration of reactants and products, the presence of a catalyst, and the activation energy of the reaction. The rate constant is related to the activation energy through the Arrhenius equation, which is given by k = A * exp(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. The rate law of a chemical reaction describes the mathematical relationship between the rate of the reaction and the concentrations of the reactants. It is determined experimentally by measuring the initial rate of the reaction under different concentrations of the reactants. The rate law is typically expressed as Rate = k[A]^m[B]^n, where k is the rate constant, A and B are the concentrations of reactants, and m and n are the orders of the reaction with respect to A and B, respectively.
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4 less than the product of 7 and a number.
Answer:
7n - 4
Step-by-step explanation:
Need this for a lesson
Answer:
median = 95
Step-by-step explanation:
the median is indicated by the line inside the box.
each interval on the number line is 5 units.
median = 75 + 4 units = 75 + 20 = 95
The following is a WFF:
(S~T) V (~U.W)
Select one:
True or
False
The given statement about the disjunction is true.
What is Disjunction?A compound statement with two distinct statements (disjuncts) connected by the wedge symbol (v) is called Disjunction.
Given is the statement -
(S~T) v (~U.W)
The given statement about the disjunction is true.
Therefore, the given statement about the disjunction is true.
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The best fit line is given by the equation y = 0.467x + 0.417 where y represents the distance in miles, and x represents the time for the trip in minutes. Use the best fit line to estimate the time for a trip that is 6 miles long. Show your reasoning.
Answer:
12 minutes
Step-by-step explanation:
To estimate the time for a trip that is 6 miles long, we need to solve for x in the equation y = 0.467x + 0.417, where y represents the distance in miles.
First, we substitute the value of y, which is 6:
6 = 0.467x + 0.417
Next, we subtract 0.417 from both sides:
5.583 = 0.467x
Finally, we divide both sides by 0.467:
x = 12
So, the estimated time for a trip that is 6 miles long is 12 minutes.
Which graph represents a linear function
Answer:
Any line except a vertical line represents a linear function.
Step-by-step explanation:
Any "diagonal" line is a linear function. Even a horizontal line represents a linear function. Only a vertical line does NOT represent a linear function.
Can anyone give me the answer ? will give brainliest
In given lines, y = -1/4 x + 7 represents a transversal line.
What is Transversal Line:A transversal is a line that, at two different locations, cuts through two lines in the same plane. Various types of angles in pairs, including successive internal angles, corresponding angles, and alternate angles, are produced by a transversal intersection with two lines.
Here we have
y = 1/6x - 3 ---- (1)
y = -1/4 x + 7 ---- (2)
y = 1/6x - 9 ---- (3)
Here given lines are in y = mx + c form
The slope of the line (1) is 1/6
The slope of the line (2) is -1/4
The slope of the line (3) is 1/6
As we can see that line (1) and line (3) have the same slope,
Hence, Lines (1) and (3) are two parallel lines
where Line (2) is transversal which cuts the given two parallel lines
Therefore,
In given lines, y = -1/4 x + 7 represents a transversal line.
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Suppose Q and R are independent events. Find P(Q and R).
P(Q) = 0.37, P(R) = 0.24
Events Q and R are not mutually exclusive, thus we are given two probabilities, P(Q) = 0.41 and P(R) = 0.44.
What is meant by probability?Probability, in its simplest form, gauges the likelihood that something will happen. When we are unsure of how an event will turn out, we might talk about the likelihood of different outcomes.Statistics is the study of events subject to probability. The probability is typically expressed as a ratio between the number of positive outcomes and all of the outcomes in the sample space. It is written as follows: Probability of an Event P(E) = (Number of Favorable Outcomes) (Sample space).There are four primary categories of probability: axiomatic, axiomatic, classical, and empirical. Probability is the discipline of mathematics that deals with the occurrence of a random event.Therefore,
Events Q and R are not mutually exclusive, thus we are given two probabilities, P(Q) = 0.41 and P(R) = 0.44.. In this instance, the likelihood of Q and R is equal to P(Q) × P(R), which is 0.1804. The solution to this issue is this.
The complete question is:
Suppose Q and R are independent events. Find the probability of Q and R when P(Q)=0.41 Ans P(R)=0.44.
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There are 2.75 ounces of cream cheese in one slice of cheesecake. How
many ounces are there in a cake with 8 slices?
A. 22 ounces
B. 21.6 ounces
C. 16 ounces
D. 10.75 ounces
Answer:
Step-by-step explanation:
2.75 x 8 = 22
a researcher records the change in weight (gain or lost) during the first semester of college for each individual in a sample of 25 freshmen, and calculates the average change in weight. this average is an example of a . group of answer choices
The average change in weight calculated by the researcher for the sample of 25 freshmen is an example of a statistic.
A statistic is a numerical summary of a sample that is used to make inferences about a population. In this case, the sample is the 25 freshmen, and the population could be all freshmen entering college.
The average change in weight is a statistic because it summarizes the data collected from the sample, and is used to make conclusions about the population.
It represents the typical or average amount of weight gained or lost by the freshmen in the sample during their first semester in college.
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The degree of the term with the greatest degree is called?
The degree of the term with the greatest degree is called the degree of the polynomial.
Here, it is necessary to remember that, by the definition, we can find the "Degree of a polynomial" by identifying the greatest degree of its terms.
The "Leading term" of a polynomial is defined mostly as the term with the highest power of the variable of the polynomial.
The numbers located in front to the variable (those numbers that multiply the variable) are called "Coefficients". The coefficient of the Leading term is known as the "Leading coefficient."
For example, given the following polynomial:
18x² + 2x - 6
You can identify that, in this case:
degree of the polynomial: 2
leading term: 18x²
leading coefficient: 18
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so what is 24 water bottles and 12 oz and 30 guest and 1 cup
There will be enough water for all the guests.
What are Measurements?Measurement is the method of comparing the properties of a quantity or object using a standard quantity.
Measurement is essential to determine the quantity of any object.
Total number of bottles = 24
Capacity of each bottle = 12 oz
Total capacity of the 24 water bottles = 12 × 24 = 288 oz
Now we have,
8 ounces = 1 cup
288 ounces = 288 / 8 = 36 cups
So there are 36 cups of water in total.
Number of guests = 30
If each guest drinks one cup of water, there will be 36 - 30 = 6 cups of water remaining.
Hence there will be enough water for the guests with 6 cups of water remaining.
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Your question is incomplete. Probably the correct question is as follows.
Before a party, Ashley bought a tray of 24 water bottles, each bottle has a capacity of 12 oz. There are 30 guests. If each guest drinks I cup of water, is there enough water for all the guests?
A direct variation function contains the points (-9, -3) and (-12, -4). Which equation represents the function?
Answer:
=3y - x =0
Step-by-step explanation:
[tex] \frac{y - y1}{x - x1} = m[/tex]
Therefore the gradient ( m ) is not given
from the question point =(-9,-3) and (-12,-4).
Gradient (m) =
[tex] \frac{y2 - y1}{x2 - x1} \\ = \frac{ - 4 - ( - 3)}{ - 12 - ( - 9)} \\ = \frac{ - 4 + 3}{ - 12 + 9} \\ = \frac{ - 1}{ - 3} \\ = \frac{1}{3} [/tex]
therefore m = 1/3.
The equation
[tex] = \frac{y - y1}{x - x1} = m \\ \frac{y - ( - 3)}{x - ( - 9)} = \frac{1}{3} \\ \frac{y + 3}{x + 9} = \frac{1}{3} \\ 3(y + 3) = 1(x + 9) \\ 3y + 9 = x + 9 \\ 3y = x + 9 - 9 \\ 3y = x + 0 \\ 3y - x = 0[/tex]
therefore the equation of the line = 3y-x=0
a spherical balloon is being inflated at 3 cubic meters per second. how fast is the surface area changing in square meters per second when the volume is 36 cubic meters?
The surface area is changing at a rate of approximately 2.94 square meters per second when the volume is 36 cubic meters and the balloon is being inflated at 3 cubic meters per second.
Volume of a sphere in terms of its radius = V = (4/3)π[tex]r^3\\[/tex] .......(1)
Differentiating on both sides, [tex]\frac{dV}{dt}[/tex] = 4π[tex]r^{2}[/tex][tex]\frac{dr}{dt}[/tex] .......(2)
r = radius of the sphere.
Spherical balloon is being inflated at a rate of 3 cubic meters per second.
[tex]\frac{dV}{dt}[/tex] = 3 [tex]m^3/sec[/tex]
Volume is 36 cubic meters.
V = 36 [tex]m^3[/tex]
From equation (1), [tex]r^3[/tex] = [tex]\frac{3V}{4\pi }[/tex]
[tex]r^3[/tex] = [tex]\frac{3*36}{4*\pi }[/tex]
[tex]r^3\\[/tex] = [tex]8.59^1^/^3[/tex]
r = 2.04 meters
Substituting this in equation (2), [tex]\frac{dr}{dt}[/tex] = [tex]\frac{dV}{dt}[/tex] / 4π[tex]r^{2}[/tex]
[tex]\frac{dr}{dt}[/tex] = 3/(4π[tex]2.04^{2}[/tex])
[tex]\frac{dr}{dt}[/tex] = 0.05737 m/sec
Now, surface area of sphere = A = 4π[tex]r^{2}[/tex]
Differentiating on both sides,
Rate of change of surface area [tex]\frac{dA}{dt}[/tex] = [tex]8\pi r\frac{dr}{dt}[/tex]
= 8* 3.14* 2.04* 0.05737
= 2.94 [tex]m^2/sec\\[/tex]
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( Which of these shapes have the same area?
A
B and C
A and C
A and B
The shapes with the same areas are (c) A and B
How to determine the shapesThe complete question is added as an attachment
From the question, we have the following parameters that can be used in our computation:
Figures A, B and C
For the figures A, B and C to have the same area, the number of boxes on the shapes must be the same
Using the above as a guide, we have the following:
Figure A = 16 square unitsFigure B = 16 square unitsFigure C = 25 square unitsBy comparison, we have
A and B = 16 square units
Hence, the shapes A and B have the same area
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what is 1/2 d =52 so what is the answer
Answer:104
Step-by-step explanation:
If 1/2 d=52,
d/2=52
Multiply both sides by 2.
Thus, d=52×2= 104.
Hope this helps! :)
A spinner is divided into five colored sections that are not of equal size: red, blue, green, yellow, and purple. The spinner is spun several times, and the results are recorded below:
Spinner Results
Color Frequency
Red 4
Blue 3
Green 2
Yellow 2
Purple 6
Based on these results, express the probability that the next spin will land on blue as a fraction in simplest form.
Answer: 3/17 or 17.64%
Step-by-step explanation:
Just add everything (adds up to 17), then you put the color frequency number as the numerator of the fraction, then have the added up number as the denominator.
The lengths of the three sides of a triangle (in inches) are consecutive integers. If the
perimeter is 27 inches, find the value of the shortest of the three side lengths.
The value of the shortest of the three side lengths is 3 inches.
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
The lengths of the three sides of a triangle (in inches) are consecutive integers.
The three sides are x, x+1 and x+2 which are consecutive.
We know that perimeter of a triangle is the sum of the three sides of a triangle.
Add the three Consecutive sides to find the perimeter.
Perimeter of triangle=x+x+1+x+2
The perimeter of triangle is 27 inhces.
27=3x+3
Substitute 3 from both sides
24=3x
Divide by 3 on both sides
8=x
Hence, the value of the shortest of the three side lengths is 3 inches.
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What is the average rate of change of the function -4≤x≤-3
Check the picture below.
[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ f(x) \qquad \begin{cases} x_1=-4\\ x_2=-3 \end{cases}\implies \cfrac{f(-3)-f(-4)}{-3 - (-4)}\implies \cfrac{[-6]~~ - ~~[-4]}{-3+4} \\\\\\ \cfrac{-6~~ + ~~4}{1}\implies \text{\LARGE -2}[/tex]
Answer:
-2
[tex]\hrulefill[/tex]
Step-by-step explanation:
The average rate of change of a function f(x) on the interval a ≤ x ≤ b can be calculated using the formula:
[tex]\boxed{\textsf{Average rate of change}=\dfrac{f(b)-f(a)}{b-a}}[/tex]
The given interval is -4 ≤ x ≤ -3, so:
[tex]a=-4[/tex][tex]b=-3[/tex]From observation of the given graph, the values of f(a) and f(b) are:
[tex]f(a)=f(-4)=-4[/tex]
[tex]f(b)=f(-3)=-6[/tex]
Substitute the values of a, b, f(a) and f(b) into the formula to calculate the average rate of change of the function f(x) on the interval -4 ≤ x ≤ -3:
[tex]\begin{aligned}\textsf{Average rate of change}&=\dfrac{f(-3)-f(-4)}{-3-(-4)}\\\\&=\dfrac{-6-(-4)}{-3-(-4)}\\\\&=\dfrac{-6+4}{-3+4}\\\\&=\dfrac{-2}{1}\\\\&=-2\end{aligned}[/tex]
Therefore, the average rate of change of the function f(x) on the given interval is -2.
The length of a rectangle is shown below:
On a coordinate grid from negative 6 to positive 6 on the x-axis and on the y-axis, two points A and B are shown. The point A is on ordered pair 1, 2, and the point B is on ordered pair negative 2, 2. A straight line joins the points A and B.
If the area of the rectangle to be drawn is 12 square units, where should points C and D be located, if they lie vertically below the line that connects B and A, to make this rectangle? (5 points)
C(−2, −1), D(1, −1)
C(−2, −4), D(1, −4)
C(−2, −2), D(1, −2)
C(−2, −5), D(1, −5
I NEEP HELP WITH THIS ONE LISTED BELOW:A solid is composed of squares and equilateral triangles. Its net is shown below:
A triangular prism and its net are shown. The net is three squares with side length 4 units. Attached above and below the middle square are equal sized triangles.
The area of each triangle is 7 square units. The surface area of the triangular prism is
square units. (Input whole number only.) (5 points)
PLEASE
The surface area of the triangular prism is 62 square units.
What is surface area?The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object is the space taken up by its outer surface.
For each three-dimensional geometrical shape, surface area and volume are determined. The area or region that an object's surface occupies is known as its surface area. Volume, on the other hand, refers to how much room an object has.
The side length of each of the three squares is 4 units.
The area of each would be:
4² = 16 sq. units.
For the three squares we have:
16(3) = 48 sq. units.
The area of each triangle is 7 sq. units.
This means together the area of the two triangles is 2(7) = 14 sq. units.
The surface area of the prism is:
14 + 48 = 62 sq. units.
Hence, the surface area of the triangular prism is 62 square units.
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shelly spent 40 minutes jogging and 22 minutes cycling and burned 620 calories. the next day, shelly swapped times, doing 22 minutes of jogging and 40 minutes of cycling and burned the same number of calories. how many calories were burned for each minute of jogging and how many for each minute of cycling?
Calories were burned for each minute of jogging is 10 calories and for each minute of cycling is 10 calories.
Two or more algebraic equations that share variables, such as x and y, are referred to be simultaneous equations.
When the equations are solved simultaneously, they are known as simultaneous equations.
Jogging be considered as x
Cycling be considered as y
Shelly spent 40 minutes jogging and 22 minutes cycling, and burned 620 calories to write this in equation we get,
40x + 22y = 620
Shelly swapped times,
22 minutes of jogging and 40 minutes of cycling and burned the same number of calories,
22x + 40y = 620
By multiplying two equation we get,
multiply equation 1 by 22 and equation 2 by 40 we get,
880x + 484y = 13640
880x + 1600y = 24800
Now subtracting equation 1 from 2 we get,
1116y = 11160
y = 11160/1116
y = 10
Therefore, putting value of y in equation 1 we get,
880x + 484(10) = 13640
880x = 13640 - 4840
x = 8800/880
x = 10
Therefore, calories were burned for each minute of jogging is 10 calories and for each minute of cycling is 10 calories.
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hized, accurate, and has a circled answer.
Problem #6 Kiran walks up to a tank of water that can hold up to 15 gallons. When it is active, a drain
empties water from the tank at a constant rate. When Kiran first sees the tank, it contains 12 gallons
of water. Four minutes later, the tank contains 10 gallons of water.
a.
b.
C.
At what rate is the amount of water in the tank changing? Use a signed number (positive or
negative) in your answer.
How many more minutes will it take for the tank to drain completely?
How many minutes before Kiran arrived was the water tank completely full?
(a) The rate at which the amount of water is changing is -1/2.
(b) It will take 20 more minutes to drain the water completely.
(c) The tank was completely full 3 minutes before Kiran arrive.
What is Unit Rate?Unit rate is the amount of one quantity for a unit amount of the other quantity.
(a) Given that,
When Kiran first sees the tank, it contains 12 gallons of water. Four minutes later, the tank contains 10 gallons of water.
Rate at which the amount of water is changing = (10 - 12) / 4 = -2/4 = -1/2 gallons per minute.
(b) Time taken to drain 1/2 gallon = 1 minute
There are 10 gallons of water remaining in the tank.
Time taken to drain 10 gallons of water = (1 ÷ 1/2) × 10 = 20 minutes
It will take 20 more minutes to drain the water completely.
(c) When Kiran arrived, water tank has 12 gallons of water.
Total amount of water in the tank is 15 gallons.
Tank has lost 3 gallons of water before Kiran arrived.
Time taken to drain 3 gallons = (1 ÷ 1/2) × 3 = 6 minutes
So the tank was completely full 3 minutes before Kiran arrive.
Hence the rate is -1/2 gallons per minute.
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Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds she can ensure that the distance between her and the rabbit is at most 100?
It is not always possible for the hunter to ensure that the distance between her and the rabbit is at most 100 after $10⁹$ rounds. The distance between the hunter and the rabbit can increase by a factor of 2 in each round, making it impossible for the hunter to catch the rabbit in some scenarios.
It is not always possible for the hunter to ensure that the distance between her and the rabbit is at most 100 after $10⁹$ rounds, regardless of how the rabbit moves or what points are reported by the tracking device. This is because the distance between the hunter and the rabbit can increase by a factor of 2 in each round, which means that after $10⁹$ rounds, the distance between them could be as much as $2⁽¹⁰⁾⁹$ times the initial distance. This value could be much larger than 100, making it impossible for the hunter to catch the rabbit. Therefore, there are scenarios in which the hunter cannot catch the rabbit within $10⁹$ rounds.
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