Smart takes the bus to work. Unfortunately, he forgets
his lunch at home. Clever grabs the lunch and drives
after the bus. The bus has a 15-minute head start and
is traveling at an average rate of 40 mph. Clever is
traveling at an average rate of 60 mph. How long does
Clever take to catch up to the bus? Draw a diagram to
solve the problem.

Answer: Riley will catch up with Lara after 6 seconds

Step-by-step explanation:

Let t represent the time it will take Riley to catch up with Lara. It means that after t seconds, Lara and Riley would have covered the same distance.

Distance = speed × time

​Lara can run 5 meters per second. It means that the distance covered by Lara in t seconds is 5t meters

​Mrs. Montoya decides to give Lara a head start of 12 meters since she runs more slowly. ​it means the total distance covered by Lara after t seconds is

5t + 12

Riley can run 7 meters per second. ​It means that the distance covered by Riley in t seconds is 7t meters.

Since the distance covered after t seconds is the same, then

7t = 5t + 12

7t - 5t = 12

2t = 12

t = 12/2

t = 6 seconds

Riley will have to run 42 meters to catch up to Lara.  It will take 6 seconds for Riley to catch up to Lara.

Given:

- Lara's speed [tex]\( v_L = 5 \)[/tex] meters per second

- Riley's speed [tex]\( v_R = 7 \)[/tex]meters per second

- Lara's head start [tex]\( d_{\text{head start}} = 12 \)[/tex] meters

Let's denote:

- t as the time it takes for Riley to catch up to Lara.

- d as the distance Riley needs to run to catch up to Lara.

1. Set up equations based on their speeds and the head start:

  - For Lara: [tex]\( d_L = v_L \cdot t \)[/tex]

  - For Riley: [tex]\( d_R = v_R \cdot t \)[/tex]

2. Account for Lara's head start:

  - Lara starts with a head start of 12 meters. Therefore, when Riley starts, Lara is already 12 meters ahead.

 [tex]\[ d_R = d_L + 12 \][/tex]

3. Substitute the expressions for [tex]\( d_L \)[/tex] and [tex]\( d_R \)[/tex]:

 [tex]\[ v_R \cdot t = v_L \cdot t + 12 \][/tex]

4. Solve for t:

[tex]\[ 7t = 5t + 12 \] \[ 7t - 5t = 12 \] \[ 2t = 12 \] \[ t = \frac{12}{2} \] \[ t = 6 \][/tex] seconds

5. Calculate the distance [tex]\( d_R \)[/tex] that Riley has to run:

[tex]\[ d_R = v_R \cdot t \] \[ d_R = 7 \cdot 6 \] \[ d_R = 42 \][/tex]

Answer:

1675 ft

Step-by-step explanation:

For us to calculate how close the windmill is to the fire, if we imagine it as a triangle, the distance from watch tower to the fire will be:

tan θ= opposite/adjacent

opposite=140 ft

adjacent = x

θ=3°

Thus;

tan 3 = 140/x

x=140/tan 3

x = 2,671.359 ft

Now, the distance from watch tower to windmill will be:

tan θ=opposite/ adjacent

θ=8°

opposite=140 ft

adjacent=y

thus

tan 8 = 140/y

y = 140/tan8

y=996.15 ft

Now, the distance between fire and wind mill will be:

x - y = 2671.359 - 996.15

x - y = 1675.209 ≈ 1675 ft

Answer:

1675 feet

Step-by-step explanation:

Refer to the image attached for diagrams and better explanations

Answer:

The C.I. at 95% is 185±9.9583 or C.I.[175 to 195].

Step-by-step explanation:

Given:

Mean =185 mg

Standard deviation=17.6

Total number of samples=12

To Find:

95% of Confidence interval .

Solution:

We know that formula for C.I is given by ,

C.I.=mean±Z[standard deviation/Sqrt(Total no of samples)]

So using all above given values we get ,

Here Zscore at 95% is Z=1.96

So

C.I.=185±1.96[17.6/Sqrt(12)]

=185±1.96[17.6/3.464]

=185±1.965[5.0808]

=185±9.9583

In short C.I. is given as at 95% C.I.[175 to 195].

Answer:

x=29

Step-by-step explanation:

x-15=14

Add 15 to 14

x=29

A=Player A runs; B=Player B runs=A-16

A+B=242 Substitute for B.

A+A-16=242 Add 16 to each side.

2A=258 Divide each side by 2.

A=129 ANSWER 1: Player A batted in 129 runs.

B=A-16=129-16=113 ANSWER 2: Player B batted in 113 runs

CHECK:

A+B=242

129+113=242

242=242

Final answer:

Player A batted in 118 runs and Player B batted in 104 runs during the 2008 regular season. We found this by setting up two equations based on the given information and solving for the variables representing the number of runs batted in by each player.

Explanation:

To solve this problem, let's use two variables, A for the number of runs batted in by Player A, and B for the number of runs batted in by Player B. We know that Player A led the league, and Player B had 14 fewer runs batted in, so we can write the following equation:

B = A - 14

We also know that together they brought in 222 runs, so we can write another equation:

A + B = 222

Substituting the first equation into the second gives us:

A + (A - 14) = 222

Simplifying this equation, we get:

2A - 14 = 222

2A = 236

A = 118

Using the value of A, we can find B:

B = 118 - 14

B = 104

Therefore, Player A batted in 118 runs and Player B batted in 104 runs during the 2008 regular season.

Answer: 918km^2

Step-by-step explanation:

[tex]area(a): 1700km^2\\percentage(p): 5.75 = \frac{5.75}{100}=0.0575\\years(y): 8[/tex]

Let x be the new dimension of the area;

[tex]x=a-(a*p*y)[/tex]

[tex]x=(1700km^2)-(1700km^2*0.0575*8)\\x=1700km^2-782km^2\\x=918km^2[/tex]

To find the value for each expression, substitute a = 3 and b = 5 into the given expressions. The values of the expressions are: 11, 13, 24, 17, 8.

To find the value for each expression, we substitute a = 3 and b = 5 into the given expressions. The expressions are:

1. 2a + b: Substituting a = 3 and b = 5, we have 2(3) + 5 = 6 + 5 = 11.
2. 2b + a: Substituting a = 3 and b = 5, we have 2(5) + 3 = 10 + 3 = 13.
3. (a + b)3: Substituting a = 3 and b = 5, we have (3 + 5)3 = 8 × 3 = 24.
4. 4b - a: Substituting a = 3 and b = 5, we have 4(5) - 3 = 20 - 3 = 17.
5. 6a - 2b: Substituting a = 3 and b = 5, we have 6(3) - 2(5) = 18 - 10 = 8.

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Final answer:

After substituting a = 3 and b = 5 into the given expressions, we calculate their values and order them from least to greatest: 6a - 2b = 8, 2a + b = 11, 2b + a = 13, 4b - a = 17, 3(a + b) = 24.

Explanation:

We are given the variables a = 3 and b = 5 and need to find the values of five expressions, then place them in order from least to greatest. Let's calculate each expression step by step:

Now placing them in order from least to greatest, we get:

Answer:

The sum of the first 6 terms of the series is 504.

Step-by-step explanation:

Given that,

Common ratio in a geometric series is, r = 0.5

First term of the series, a = 256

We need to find the sum of the first 6 terms in the series. If a and r area the first term and common ratio of a series, then the series becomes:

[tex]a,ar^1,ar^2,ar^3......[/tex]

The sum of n terms of a GP is given by :

[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]

Here, n = 6

[tex]S_n=\dfrac{256\times (1-(0.5)^6)}{1-0.5}\\\\S_n=504[/tex]

So, the sum of the first 6 terms of the series is 504.

Answer:

A) 330 cd/sq. M per m

Step-by-step explanation:

To find the average rate at which the light intensity decreases as Caleb goes from a depth of 5m to a depth of 14m inside the cave, we need to calculate the change in light intensity and divide it by the change in depth.

To find the average rate at which the light intensity decreases as Caleb goes from a depth of 5m to a depth of 14m, we need to calculate the change in light intensity and divide it by the change in depth.

Let's denote the initial light intensity at 5m as I1 and the final light intensity at 14m as I2. Using the graph, we can estimate the values of I1 and I2.

Then, the average rate of decrease in light intensity is given by:

Average rate = (I2 - I1) / (14m - 5m)

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Answer:

The correct answer is must have a positive slope.

Step-by-step explanation:

The coefficient of determination varies between -1 to 1. It shows the how strong is the relationship between two variables.

Coefficient closer to -1 indicate negative relationship and that y decreases with increase in x and the regression equation has a negative slope.

Coefficient closer to 1 indicate positive relationship and that y increases with increase in x and the regression equation has a positive slope.

Coefficient closer to 0 indicate that there is no possible relationship between the variables under consideration. It is not possible to construct a particular regression equation.

Answer:

60

Step-by-step explanation:

Answer:

10 cubes on each side

Step-by-step explanation:

The amount of cubes to make a square frame given was 36

If you start by putting ten cubes on one side then adding nine off of the the end cubes and continue that pattern you end up with 10 cubes on each side.

There will be 6 cubes on each side of the frame.

Explanation:

To understand the relationship between the number of cubes used and the length of the side of the square frame, let's first consider the given example with 12 cubes.

 For the smaller frame with 12 cubes, we know that the frame is square and has 4 cubes on each side. This means that the total number of cubes used to make the frame is the perimeter of the square minus the 4 corner cubes, which are counted twice (once for each side they belong to). The perimeter of the smaller square frame is [tex]\(4 \times 4 = 16\)[/tex] cubes, but since we subtract the 4 corner cubes, the total number of cubes needed is [tex]\(16 - 4 = 12\)[/tex] cubes.

Now, we are given that 36 cubes are used to make a larger square frame. Using the same logic, we can find the length of the side of this larger frame. Let \(n\) be the number of cubes on each side of the larger frame. The perimeter of this larger square frame would be [tex]\(4n\)[/tex] , and we would again subtract the 4 corner cubes. Therefore, the equation to find [tex]\(n\)[/tex] is:

 [tex]\[4n - 4 = 36\][/tex]

 Adding 4 to both sides gives us:

[tex]\[4n = 40\][/tex]

Dividing both sides by 4 gives us:

[tex]\[n = 10\][/tex]

So, there are 10 cubes on each side of the larger frame. However, we need to account for the fact that the frame is hollow, meaning the corner cubes are not present. Since there are 4 corners without cubes, we subtract 1 cube from each side to get the number of cubes actually used for each side:

[tex]\[n - 1 = 10 - 1 = 9\][/tex]

 Now, we can calculate the total number of cubes used for the frame with 9 cubes on each side:

[tex]\[4 \times 9 = 36\][/tex]

This confirms that with 9 cubes on each side, we indeed use 36 cubes to make the frame. However, since the question asks for the number of cubes on each side of the frame, we must consider that the 36 cubes include the 4 corner cubes that are counted twice (once for each side). Therefore, we add back the 4 corner cubes to the length of each side:

[tex]\[9 + 1 = 10\][/tex]

 Thus, there are 10 cubes on each side of the frame, but since we are considering the outer dimension of the frame (including the corners), we have to add the corner cubes that are shared by two sides. Since there are 4 corners, and each corner contributes 1 cube to 2 sides, we divide by 2 to avoid double counting:

[tex]\[10 + \frac{4}{2} = 10 + 2 = 12\][/tex]

 However, this calculation is incorrect because we've already accounted for the corner cubes when we subtracted 4 from the perimeter to get the number of cubes on each side. The correct number of cubes on each side is 9, as calculated previously. Since the question asks for the number of cubes on each side of the frame, not the length of the side including the corners, the correct answer is 9.

Upon reviewing the solution, it is clear that there was an error in the final step. The correct number of cubes on each side of the frame is indeed 9, not 12. Therefore, the final answer should be:

There will be 9 cubes on each side of the frame.

Answer:

21 inches

Step-by-step explanation:

In 1 rotation the angle at the centre = 360°

For 120° the fraction turned = [tex]\frac{120}{360}[/tex] = [tex]\frac{1}{3}[/tex], thus

distance travelled = [tex]\frac{1}{3}[/tex] × 63 = 21 inches

If the wheel rotates only 120° it travel 21 inches.

The angle at the center of one rotation is 360°.

The fraction for 120° changed[tex]=\frac{120}{360} =\frac{1}{3}[/tex]

Total 63 inches in one complete rotation of a wheel.

Thus distance travelled[tex]=\frac{1}{3}\times 63= 21[/tex]

The wheel travel on the bicycle is 21 inches.

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Step-by-step explanation:

42° + x = 115° (alternate angles)

x = 115° - 42°

x = 73°

y = 180° - (115° + 42°) (by angle sum property of triangle)

y = 180° - 167°

y = 13°

z = 42° (corresponding angles)

Answer:

b. y = -7x + 4

   1.    y = -7(-5)+ 4 = 35+4= 39

   2. -24= -7x + 4

    -28 = -7x

       4 = x

    3. y = -7(0)+4= 4

    4. 0 = -7x+4

        -4= -7x

         4/7 = x

c. 3y - 5x = 15

 1. 3y - 5(6)

    3y-30=15

    3y=45

      y = 15

   2. 3(-10)-5x=15

        -30-5x=15

         -5x = 45

            x = -9

    3. 3y-5(0)=15

        3y=15

         y=5

    4. 3(0)-5x=15

         0-5x=15

          -5x=15

           x=-3

Step-by-step explanation:

Answer: Given:

(r, θ) is equivalent to (x, y) = (7, 5).

By definition,

r = √(7² + 5²) = 8.6023

θ = tan⁻¹ (7/5) = 0.9505 rad

Therefore

2r = 17.2047

θ + π/2 = 0.9505 + π/2 = 2.5213

In rectangular coordinates,

x = 2r cos(θ + π/2) = 17.2047*cos(2.5213) = -14

y = 2r sin(θ + π/2) = 17.2047*sin(2.5213) = 10

Answer: (-14, 10)

The rectangular coordinates of the polar coordinates (2√2, -π/12) are (-14,10).

The polar coordinate system is a two-dimensional coordinate system in mathematics in which points are defined by an angle and a distance from a central point known as the pole (equivalent to the origin in the more familiar Cartesian coordinate system).

We know polar coordinates are in the form (r, θ).

Also, (r, θ) = (x, y) = (7, 5).

So, r = √(7² + 5²)

r = 8.6023

and, θ = tan⁻¹ (7/5)

θ = 0.9505 rad

Now, 2r = 17.2047

and, θ + π/2 = 0.9505 + π/2 = 2.5213

Thus, the rectangular coordinates are

x = 2r cos(θ + π/2)

x = 17.2047 . cos(2.5213)

x = -14

and, y = 2r sin(θ + π/2)

y = 17.2047 . sin(2.5213)

y = 10

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Answer:

The area of the larger octagon is  [tex]236.18\ in^2[/tex]

Step-by-step explanation:

The question is

What is the area of the larger octagon?

we know that

If two figures are similar, then the ratio of its areas is equal to the scale factor squared

so

Let

z ----> the scale factor

x ----> the area of the larger octagon

y ---> the area of the smaller octagon

[tex]z^2=\frac{x}{y}[/tex]

we have that

[tex]z=3.5[/tex]

Because, in similar figures the ratio of corresponding sides is proportional and this ratio is equal to the scale factor

we have

[tex]z=3.5[/tex]

[tex]y=19.28\ in^2[/tex]

substitute the given values

[tex]3.5^2=\frac{x}{19.28}[/tex]

[tex]x=12.25(19.28)=236.18\ in^2[/tex]

Step-by-step explanation:

Given

Radius (r) = 9 yards

Area of circle

= πr²

= 3.14 * 9²

= 3.14 * 81

= 254.34 yards²

Answer: the winnings are reduced

Step-by-step explanation:

Given 10 percent chance to win $1,000 for $100. That is

Gain = $900

Assuming diminishing marginal utility of dollars, when the utility of the gain and the money used for bet are considered, it is discovered that the utility of the $100 used to make the bet is greater than the $900 that you might gain if you win the bet.

this is not a fair bet in terms of utility because the winnings are reduced.

A mathematically fair bet does not account for the diminishing marginal utility of money. The expected utility calculated with a utility function might show a different picture, revealing the bet might not be fair in terms of utility. Samuelson's story illustrates statistical independence and that repeated gambles with positive expectations lead to better outcomes.

A "mathematically fair bet" is one where the expected value of the bet is equal to the cost of making the bet. In other words, if you bet $100 for a 10 percent chance to win $1,000, the mathematically calculated expectation would be $100, or 0.10 times $1,000. However, the concept of diminishing marginal utility suggests that the utility or satisfaction derived from each additional dollar decreases as one has more money. Thus, the utility of the potential win cannot simply be calculated by multiplying the probabilities by the monetary outcome.

Under the concept of expected utility, the expected utility of a bet should also be considered. To calculate this, one might use a utility function, like the von Neumann-Morgenstern utility function, to determine the utility of each possible outcome and then finding the average utility. For example, let's assume that the utility function is represented by the square root of the dollar amount (this is just an example, utility functions can take many forms). Hence, the expected utility of winning $1,000 would be the square root of $1,000, not just $1,000.

Paul Samuelson's story illustrates the idea of statistical independence and the law of large numbers, where repeated play of a gamble with positive expectation can lead to more stable and predictable results, as opposed to a single high-risk gamble. This idea reflects human behavior and the role of hope in encouraging smart risk-taking, even when each individual bet may not be favorable in terms of utility.

The given expression 2(x − 7) + 10 matches with two times the difference of x and seven plus ten. Option B is correct.

Given that,
Which phrase matches the algebraic expression 2(x − 7) + 10 is to be determined.

In mathematics, it deals with numbers of operations according to the statements.

The procedure in mathematics to utilize and analyze the function or expression to make the function or expression uncomplicated or more coherent is called simplifying and the process is called simplification.

Here,
2(x − 7) + 10
When we turn the above algebraic expression into word phrase,

"two times the difference of x and seven plus ten"

Thus, the given algebraic expression 2(x − 7) + 10 matches with two times the difference of x and seven plus ten. Option B is correct.

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Final answer:

To solve for x when g(x) = 0 for the function g(x) = 5x + 5, subtract 5 from both sides and then divide by 5, resulting in x = -1.

Explanation:

To solve the equation g(x) = 5x + 5 when g(x) = 0, you need to find the value of x that makes the equation true.

Set up the equation: 5x + 5 = 0.

Subtract 5 from both sides: 5x = -5.

Divide both sides by 5 to solve for x: x = -5 / 5.

Thus, x = -1.

Therefore, the solution for x when g(x) = 0 is x = -1.

Answer:

The slope of a line graphed to represent the volume of water in the pool can be described as negative.

The slope of a line graphed to represent the volume of water in a pool over time would be described as negative

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of line be represented as A

Now , the value of A is

Let the first point be P ( 0 , 50 )

Let the second point be Q ( 5 , 20 )

Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

Slope m = ( 50 - 20 ) / ( 0 - 5 )

m = 30 / -5

m = -6

Now , the slope of the line is negative and volume of water in the pool over time decreases at a rate of 6

Hence , the slope is negative

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The complete question is attached below :

The table shows the gallons of water in a pool over time. Choose the term that describes the slope of the line of a graph representing the data in the table. The slope of a line graphed to represent the volume of water in a pool over time would be described as___

Answer:

Refer below.

Step-by-step explanation:

The principal will have to increase the number of teachers next year.

We found out that the seventh grade enrollment was 120% of the number of eight graders, which is greater than 110%.

Answer:

#3

Step-by-step explanation:

Similar to the US highway system Rome realized if they wanted to control their territory they were going to need to be able to squash resistance that wasn't in the immediate area and move resources form place to place easily

Answer:

c

Step-by-step explanation:

i took the test

Final answer:

In mathematics, when an operation involves an even number of signs, the result will be positive. This applies to multiplication and division rules, where similarly signed numbers yield a positive result. Principles of mathematical operations also extend to probabilities, where conditions and the complement of events are crucial to determining event outcomes.

Explanation:

If there is an even number of signs in a mathematical operation, such as multiplication or division, the solution will have a positive sign. This is because multiplying or dividing two positive numbers, or two negative numbers, results in a positive outcome. However, multiplying or dividing numbers with opposite signs results in a negative outcome.

For instance:

When two positive numbers multiply, the answer has a +ve sign, e.g., 2x3 = 6.

When two negative numbers multiply, the answer also has a +ve sign, e.g., (-4) x (-3) = 12.

In probability, events A and B being given conditions, such as event A given B (A GIVEN B) or event B given A (B GIVEN A), can impact the likelihood of those events occurring. The complement of an event (like the complement of A) refers to all outcomes that are not part of the event A.

Principles of Mathematical Operations and Probability

Understanding these principles helps solve many mathematical problems and assess probabilities, especially when working with equations with an unknown squared, which will typically result in two solutions, or when calculating the possibilities of a dice roll in probability exercises.

Answer:

45 Handshakes

Step-by-step explanation:

45 Handshakes have taken place all together

Answer:

45

Step-by-step explanation:

Answer:

D on e2020

Step-by-step explanation:

got it right

Answer:

D. (StartFraction 7 Over 2 EndFraction) squared

Step-by-step explanation:

Answer:

the answer is D

Step-by-step explanation:

D) Westfield's data shows greater variability, since Westfield's MAD is approximately 2.9 times greater than Eastfield's MAD.  

Answer:

D) Westfield's data shows greater variability, since Westfield's MAD is approximately 2.9 times greater than Eastfield's MAD.

Step-by-step explanation:

Westfield's data shows greater variability, since Westfield's MAD is approximately 2.9 times greater than Eastfield's MAD.

Westfield's MAD = 6.56

Eastfield's MAD = 2.28

Therefore,

6.56

2.28

= 2.877

MAD =

∑|x − X|

n

, where x = Data value, X = Mean, and n = Number of values

Answer:

The answer to your question is 566 m²

Step-by-step explanation:

Data

length = 19 m

width = 4 m

height = 9 m

Process

1.- Calculate the area of the 6 faces

-Area of the bases

Area = 19 x 4 x 2 = 152 m²

-Area of the lateral faces

Area = 4 x 9 x 2 = 72 m²

-Area of the frontal and the opposite faces

Area = 19 x 9 x 2 = 342 m²

2.- Calculate the total area

Total area = 152 + 72 + 342

                 = 566 m²