Jack and Charlie cycled from home to park along the same route at 11 km/h and 8 km/h respectively. Jack left home 1 h later than Charlie and arrived at the park 1/2 h earlier than Charlie. How long did Jack take to cycle from home to park?

The Equation of line is y= -3/2x + 60

From the graph we take two coordinates as (2, 0) and (0, 3)

We know the formula for slope

Slope= (Change in y)/ (Change in x)

Slope = (3-0)/ (0-2)

Slope= 3 / (-2)

Slope= -3/2

Now, Equation of line

y - 0 = -3/2 (x-  2)

y= -3/2x + 6

Thus, the Equation of line is y= -3/2x + 60.

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The required angle is .

Given the triangle and name it as triangle ABC. In triangle ABC, ∠C = 29 and AB =6.78, BC=4, AC = 10.

To find angle A in triangle ABC, use the Law of Cosines, which states:

[tex]c^2 = a^2 + b^2[/tex]- 2ab x cos(C)

That implies,

AB = 6.78 (side a)

BC = 4 (side b)

AC = 10 (side c)

∠C = 29°

Substituting the given values into the Law of Cosines formula, gives:

[tex]10^2 = 6.78^2 + 4^2[/tex] - 2 x 6.78 x 4 x cos(29°)

Simplifying the equation:

100 = 46.2084 + 16 - 54.24 x cos(29°)

Rearranging the equation to isolate the cosine term:

54.24 x cos(29°) = 46.2084 + 16 - 100

54.24 x cos(29°) = -37.7916

Solve for the cosine term:

cos(29°) = -37.7916 / 54.24

cos(29°) = -0.696

To find angle A,  use the inverse cosine (cos⁻¹) function:

∠A = cos⁻¹(-0.696)

Calculating the value of angle A using a calculator or trigonometric table, we find:

∠A = 133.64°

Therefore, angle A in triangle ABC is approximately 133.64°.

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The Length of the rod is 3/5 meters.

The civil engineer wants to find the length of a rod that stretches for 1 meter and can be given by the function x=2y^(3/2).

To find the length of the rod, we need to integrate the function x=2y^(3/2) with respect to y. Integrating both sides of the equation,

we have:'int dx = int 2y^(3/2) evaluating the left-hand side gives x = 2/5 y^(5/2) + C, where C is the constant of integration. To find the value of C,

we use the given information that the rod stretches for 1 meter. At y = 0, x = 0 since the rod has no length when it is not stretched. At y = 1, x = 1 since the rod stretches for 1 meter.

Therefore, we have:1 = 2/5 (1)^(5/2) + C1 = 2/5 + CC = 3/5 Substituting C = 3/5 back into the equation for x,

we have:x = 2/5 y^(5/2) + 3/5

The length of the rod is given by the value of x when y = 1. Substituting y = 1,

we have:x = 2/5 (1)^(5/2) + 3/5 = 3/5

The length of the rod is 3/5 meters.

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The nearest tenth, approximately 93.5% of Container A is full after the water is pumped into Container B.

To determine the percentage of Container A that is full after the water is pumped into Container B, we need to compare the volumes of the two containers.

The volume of a cylinder can be calculated using the formula: V = πr^2h, where V is the volume, π is a constant (approximately 3.14159), r is the radius, and h is the height.

For Container A:

Radius (r) = Diameter / 2 = 12 ft / 2 = 6 ft

Height (h) = 14 ft

For Container B:

Radius (r) = Diameter / 2 = 10 ft / 2 = 5 ft

Height (h) = 20 ft

Now, let's calculate the volumes of the two containers:

Volume of Container A = π * (6 ft)^2 * 14 ft ≈ 1,679.65 ft^3

Volume of Container B = π * (5 ft)^2 * 20 ft ≈ 1,570.8 ft^3

To find the percentage of Container A that is full, we need to calculate the ratio of the volume of water in Container B to the volume of Container A:

Ratio = Volume of Container B / Volume of Container A

Ratio = 1,570.8 ft^3 / 1,679.65 ft^3 ≈ 0.9347

Finally, to convert this ratio to a percentage, we multiply it by 100:

Percentage = Ratio * 100

Percentage ≈ 0.9347 * 100 ≈ 93.5%

Therefore, to the nearest tenth, approximately 93.5% of Container A is full after the water is pumped into Container B.

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The algorithm justifies the answer of 3/5 as the fraction each person gets.

Representation of the fractions on the second row:

From what you described, two of the loaves were divided into thirds.

This means each person receives one third, and there is one third remaining. Then, this remaining third from the second loaf was further divided into fifths.

Therefore, each person receives one fifth from this remaining third.

So, the representation of the fractions on the second row would be:

Each person receives 1/3 (one third) from the two loaves.

Each person receives 1/5 (one fifth) from the remaining third.

Algorithm/Abstract strategy to justify the 3/5 found as the answer:

To find the final answer of 3/5, we can follow the steps you provided:

Divide two loaves into thirds, giving each person 1/3.

Divide the remaining third from the second loaf into fifths, giving each person 1/5.

Combining the fractions, each person has 1/3 + 1/5.

To add these fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 3 and 5 is 15. We can convert 1/3 and 1/5 to have a denominator of 15:

1/3 = 5/15 (multiplying numerator and denominator by 5)

1/5 = 3/15 (multiplying numerator and denominator by 3)

Now, we can add the fractions:

5/15 + 3/15 = 8/15

Therefore, each person receives 8/15 of a loaf.

Simplifying this fraction, we get 3/5.

Hence, the algorithm justifies the answer of 3/5 as the fraction each person gets.

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Answer:

D

Step-by-step explanation:

the right triangle contains h , the horizontal leg and the sloping side which is the hypotenuse of the right triangle.

the horizontal leg is half the measure of the side of the square base.

horizontal leg = 8 ÷ 2 = 4

using Pythagoras' identity in the right triangle

the square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is

h² + 4² = 10² ( subtract 4² from both sides )

h² = 10² - 4² ( take square root of both sides )

h = [tex]\sqrt{10^2-4^2}[/tex]

To determine the balance of iron supplement dose and ordinary food servings needed to meet the patients' nutritional needs, let's assign variables to represent the quantities:

Let:

- x = number of iron supplement doses per day

- y = number of ordinary food servings per day

Based on the information given, we can establish the following equations:

Equation 1: Iron Balance

1.2µg * x + 0.4µg * y = 5.2µg

Equation 2: Potassium Balance

0.3mg * x + 0.1mg * y = 1.3mg

We can solve this system of equations to find the values of x and y that satisfy both equations.

Multiplying Equation 1 by 10 and Equation 2 by 1000 will help us eliminate the decimal points:

Equation 1 (revised): 12µg * x + 4µg * y = 52µg

Equation 2 (revised): 300µg * x + 100µg * y = 1300µg

Now, we can use any method to solve the equations. Let's solve them using the substitution method:

From Equation 1 (revised), we can express x in terms of y:

12µg * x = 52µg - 4µg * y

x = (52µg - 4µg * y) / 12µg

x = (13µg - µg * y) / 3µg

x = 13/3 - y/3

Substituting this value of x into Equation 2 (revised):

300µg * (13/3 - y/3) + 100µg * y = 1300µg

Simplifying and solving for y:

(3900µg - 100µg * y + 100µg * y) / 3 = 1300µg

3900µg / 3 = 1300µg

1300µg = 1300µg

The equation is satisfied for any value of y. This means that there is no unique solution for the system of equations. In other words, any combination of iron supplement doses (x) and ordinary food servings (y) that satisfy the equation 1.2µg * x + 0.4µg * y = 5.2µg will also satisfy the equation 0.3mg * x + 0.1mg * y = 1.3mg.

Therefore, there are multiple ways to achieve the balance of iron and potassium needed to meet the patients' nutritional needs. The specific values of x and y will depend on the preferences of the patients and the dosing recommendations by healthcare professionals.

The measure of angle BDA is 3

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

ABC = 4x - 2

DBC = 3x - 3

The figure is a rhombus

This means that

ABC = 2 * DBC

So, we have

4x - 2 =2 * (3x - 3)

When evaluated, we have

4x - 2 = 6x - 6

When solved for x, we have

2x = 4

So, we have

x = 2

This also means that

BDA = DBC = 3x - 3

So, we have

BDA = 3(2) - 3

Evaluate

BDA = 3

Hence, the measure of angle BDA is 3

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The solution in interval notation is; (-∞,  49/2).

Inequality is defined as the relation which makes a non-equal comparison between two given functions.

To solve the equation 5/16x - 7/4 < 3/4x + 21/2, we can simplify both sides:

5/16x - 7/4 < 3/4x + 21/2

Combining like terms:

5/16x  -3/4x  < 21/2 + 7/4

8/16x < 49/4

1/2x < 49/4

Simplifying the fraction;

x < 49/2

Therefore, the solution in interval notation is (-∞,  49/2).

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If the dimensions of your driveway are 10.2 ft. by 15.7 ft and the tarp covers your entire driveway,  the square feet are covered is [tex]160.14ft^{2}[/tex]

In mathematics, a dimension is the length or width of an area, region, or space in one direction. It is just the measurement of an object's length, width, and height.

With the given conditions,  we can formulate the expression as

;10.2 ft. * 15.7 ft

=160.14

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The values of b and c is 5 and 6.

To factorize the trinomial x² + 5x + 6, we need to find two integers whose product is 6 and whose sum is 5.

From the given table, we can see that the integers 2 and 3 satisfy these conditions.

Therefore, we can rewrite the trinomial as:

x² + 5x + 6 = x² + 2x + 3x + 6

x² + 2x + 3x + 6 = (x² + 2x) + (3x + 6)

Now, we can factor out the common terms from each group:

x² + 2x + 3x + 6 = x(x + 2) + 3(x + 2)

                          = (x + 2)(x + 3)

Therefore, the factored form of the trinomial x² + 5x + 6 is (x + 2)(x + 3).

Regarding the values of b and c, we can see that b = 5 and c = 6 in the trinomial x² + 5x + 6.

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a) Mean for the first year of the contract: $63,500

The standard deviation for the first year of the contract: $2,000.

b) Mean for the second year of the contract: $65,405.

The standard deviation for the second year of the contract: $60.

We have,

To find the mean and standard deviation for the first year of the contract, we can use the given information and the properties of the normal distribution.

Given:

The mean salary for professors in the previous year = $62,000

Standard deviation in the previous year = $2,000

Raise in the first year = $1,500

Mean for the first year of the contract:

The mean salary for the first year can be obtained by adding the raise to the previous mean:

Mean = Previous Mean + Raise

Mean = $62,000 + $1,500

Mean = $63,500

The standard deviation for the first year of the contract:

Since each professor receives the same raise, the standard deviation remains the same:

Standard Deviation = $2,000

Therefore, for the first year of the contract, the mean salary is $63,500, and the standard deviation remains $2,000.

Now,

In the second year of the contract, each professor receives a 3% raise based on their salary during the first year of the contract.

To find the mean and standard deviation for the second year, we can use the given information and the properties of the normal distribution.

Mean for the second year of the contract:

To calculate the mean for the second year, we need to add a 3% raise to the mean salary of the first year:

Mean = Mean of the first year + (3% * Mean of the first year)

Mean = $63,500 + (0.03 * $63,500)

Mean = $63,500 + $1,905

Mean = $65,405

The standard deviation for the second year of the contract:

Since each professor receives a raise based on their salary from the first year, the standard deviation also increases. To calculate the standard deviation, we multiply the standard deviation from the first year by the percentage increase:

Standard Deviation = Standard Deviation of the first year * (Percentage Increase / 100)

Standard Deviation = $2,000 * (3 / 100)

Standard Deviation = $2,000 * 0.03

Standard Deviation = $60

Therefore, for the second year of the contract, the mean salary is $65,405, and the standard deviation is $60.

Thus,

a) Mean for the first year of the contract: $63,500

The standard deviation for the first year of the contract: $2,000.

b) Mean for the second year of the contract: $65,405.

The standard deviation for the second year of the contract: $60.

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Answer:

1.452

Step-by-step explanation:

1.45169 rounded to the thousandths place would be 1.452

Answer:

8(3+4p)

Step-by-step explanation:

8 goes into 24 and 32, so it can be factored out:

[tex]24+32p=8(3+4p)[/tex]

Answer:

8(3 + 4p)

Step-by-step explanation:

Factor 24 + 32p

First, factor out the GCF. In this case, the GCF is 8, and we have

8(3 + 4p)

We can't factor anymore so the answer is 8(3 + 4p)

Answer:

To write the result in lowest terms, we need to simplify the fractions by dividing both the numerator and the denominator by their greatest common factor (GCF). Here are the solutions for each problem:

1.) -15-(5)= -20. This is already in lowest terms because it is an integer.

2.) 5/9 divided by 10/18= (5/9) * (18/10) = 90/90 = 1. This is in lowest terms because the GCF of 90 and 90 is 90.

3.) 2/5+4/7= (14/35)+(20/35) = 34/35. This is in lowest terms because the GCF of 34 and 35 is 1.

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Pat's approximate monthly Payment would be $304.32 to repay all her loans after the deferment period.

The Pat's monthly payment, we need to consider the terms and interest rates of each loan. Let's break down the calculation step by step:

1. Federally subsidized student loans:

  - Pat receives four annual loans, each for $5400 at an interest rate of 6.5%.

  - Since the loans are annual, we need to calculate the interest for each year and add it to the principal amount.

  - The total principal amount for the four loans is $5400 * 4 = $21,600.

  - The interest for each year is $21,600 * 6.5% = $1,404.

  - Therefore, the total amount owed for the federally subsidized loans is $21,600 + ($1,404 * 4) = $27,816.

2. Private student loan:

  - Pat takes out a private student loan for $4100 at an interest rate of 7.5% for a term of 10 years.

  - The loan is capitalized, which means the interest is added to the principal amount.

  - The total principal amount for the private loan is $4100.

  - The interest for each year is $4100 * 7.5% = $307.50.

  - Since Pat capitalizes the interest for her last year of college, the loan will accrue interest for a total of 9 months.

  - Therefore, the total interest accrued for the private loan is ($307.50 * 9) = $2767.50.

  - The total amount owed for the private loan is $4100 + $2767.50 = $6867.50.

3. Total amount owed:

  - To find the total amount owed by Pat, we add the amounts from the federally subsidized loans and the private loan.

  - Total amount owed = $27,816 + $6867.50 = $34,683.50.

4. Monthly payment:

  - The monthly payment is calculated based on the total amount owed and the repayment term.

  - The term is 10 years for the private loan, but since payments are deferred until 6 months after graduation, the actual term is 10 years - 0.5 years = 9.5 years.

  - The number of monthly payments is 9.5 years * 12 months/year = 114 months.

  - Therefore, the monthly payment is $34,683.50 / 114 months ≈ $304.32.

So, Pat's approximate monthly payment would be $304.32 to repay all her loans after the deferment period.

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Answer:

------------------------

Direct variation equation in terms of given values:

10 cm³ of oil has a mass of 9 grams:

Substitute values of m and v and find the value of k:

Substitute the value of k back to initial equation:

The matching choice is B.

Answer:

[tex]m= \frac{9}{10}*v[/tex]

Step-by-step explanation:

Since the mass of cooking oil is directly proportional to the oil's volume, we can write the following equation:

m = kv

where k is the constant of proportionality.

We know that when v = 10, m = 9, so we can plug these values into the equation to solve for k:

9 = k * 10

k =[tex]\frac{9}{10}[/tex]

Now, we can plug k =[tex]\frac{9}{10}[/tex] into the original equation to get the following equation:

m =[tex]\frac{9}{10}[/tex] v

Answer:

12

Step-by-step explanation:

Based on the functions given, it should be noted that the values will be 2, -2x and -8x.

Using the definition of the derivative, we have:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

= lim(h->0) [2(x + h) - 2x] / h

= lim(h->0) 2h / h

= lim(h->0) 2

= 2

Therefore, f'(x) = 2.

For f(x) = 9 - x²:

Using the definition of the derivative, we have:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

= lim(h->0) [9 - (x + h)² - (9 - x²)] / h

= lim(h->0) [9 - (x² + 2xh + h²) - 9 + x²] / h

= lim(h->0) [-2xh - h²] / h

= lim(h->0) (-2x - h)

= -2x

Therefore, f'(x) = -2x.

For f(x) = -4x²:

Using the definition of the derivative, we have:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

= lim(h->0) [-4(x + h)² - (-4x²)] / h

= lim(h->0) [-4(x² + 2xh + h²) + 4x²] / h

= lim(h->0) [-4x² - 8xh - 4h² + 4x²] / h

= lim(h->0) [-8xh - 4h²] / h

= lim(h->0) (-8x - 4h)

= -8x

Therefore, f'(x) = -8x.

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Answer:

B) Subtract 6 from both sides of the inequality.

Step-by-step explanation:

By doing so, we have the following:

x/2 + 6 > 42

x/2 + 6 > 42 - 6

x/2 > 36

(x/2)*2 > 36*2

x > 72

Answer:

B. subtract 6 from both sides of the inequality.

Step-by-step explanation:

Kendall, when solving inequality, would isolate the variable, x.

The first step will be to subtract 6 from both sides of the inequality:

[tex]\frac{x}{2} + 6 > 42\\\frac{x}{2} + 6 (-6) > 42 (-6)\\\frac{x}{2} > 36[/tex]

The next step will be to multiply 2 to both sides of the inequality:

[tex]\frac{x}{2} > 36\\\frac{x}{2} *2 > 36 *2\\x > 36 * 2\\x > 72[/tex]

x > 72 would be the answer.

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Answer:

.35x = 140(.15)

.35x = 21

x = 60 oz of 35% salt.

The scientist will need 60 oz of the 35% salt solution and 80 oz of water.

Answer:

Let x be the measure of the spacing between the bars.

6.25" + 5x = 36"

5x = 29.75"

x = 5.95"

Answer:

Step-by-step explanation:

To find the volume of the storage bin, we need to calculate the volumes of both the cylindrical portion and the conical top, and then add them together.

The volume of the cylindrical portion can be calculated using the formula:

V_cylinder = π * r^2 * h

where r is the radius and h is the height of the cylindrical portion.

Substituting the given values, we have:

V_cylinder = π * (4.9 ft)^2 * 9.7 ftV_cylinder ≈ 748.07 ft³ (rounded to two decimal places)

The volume of the conical top can be calculated using the formula:

V_cone = (1/3) * π * r^2 * H_cone

where r is the radius and H_cone is the height of the conical top.

The height of the conical top can be obtained by subtracting the height of the cylindrical portion from the overall height:

H_cone = H - h = 16.3 ft - 9.7 ft = 6.6 ft

Substituting the given values, we have:

V_cone = (1/3) * π * (4.9 ft)^2 * 6.6 ftV_cone ≈ 243.24 ft³ (rounded to two decimal places)

To find the total volume, we add the volume of the cylindrical portion and the volume of the conical top:

Total volume = V_cylinder + V_cone

Total volume ≈ 748.07 ft³ + 243.24 ft³

Total volume ≈ 991.31 ft³ (rounded to one decimal place)

Therefore, the volume of the storage bin is approximately 991.3 ft³ (rounded to the nearest tenth).

Thus the required volume is, 975.05  ft³

Given that,

radius = r = 4.9

Height of cylindrical potion = h = 9.7

Overall height = 16.3

Since,

total height = Height of the cylinder + height of the cone

Height of the cone = 16.3 - 9.7

                                = 6.6 m

Since we know that,

Volume of a cylinder = πr² h

⇒ π (4.9)²(9.7)

⇒ 731.29 ft³

Since we also know that

Volume of a cone = (1/3)πr² h

= 731.29/3

= 243.76  ft³

Volume of the bin = volume of cone + volume of cylinder

= 731.29 ft³  + 243.76  ft³

Hence the volume be,

= 975.05  ft³

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Answer:

1 < m < 4

Step-by-step explanation:

If the roots of function f(x) are not real, then the discriminant (the part under the square root sign) will be negative.

Set the discriminant less than zero and rewrite in standard form:

[tex]\begin{aligned}16-4m(-m+5)& < 0\\16+4m^2-20m& < 0\\4m^2-20m+16& < 0\\4(m^2-5m+4)& < 0\\m^2-5m+4& < 0\end{aligned}[/tex]

Factor the quadratic:

[tex]\begin{aligned}m^2-5m+4& < 0\\m^2-4m-m+4& < 0\\m(m-4)-1(m-4)& < 0\\(m-1)(m-4)& < 0\end{aligned}[/tex]

The leading coefficient of the quadratic m² - 5m + 4 is positive.

Therefore, the graph will be a parabola that opens upwards.

This means that the interval where the parabola is below the x-axis (negative) is between the zeros of the quadratic. Since the zeros are m = 1 and m = 4, the solution to the inequality is 1 < m < 4.

Therefore, the values of m for which the roots of function f(x) will be non-real are 1 < m < 4.

Edwin needs to sell 44,625 jars of jam to break even.

To determine how many jars of jam Edwin needs to sell to break even, we'll calculate the breakeven point using the following formula:

Breakeven Point = Fixed Expenses / (Selling Price per Unit - Variable Cost per Unit)

Given information:

Selling Price per Unit (SP) = $1.90

Variable Cost per Unit (VC) = $1.10

Fixed Expenses = $35,700.00 per year

Plugging in the values into the formula:

Breakeven Point = $35,700 / ($1.90 - $1.10)

Breakeven Point = $35,700 / $0.80

Breakeven Point = 44,625 jars

Therefore, Edwin needs to sell 44,625 jars of jam to break even.

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Answer:

880 feet

Step-by-step explanation:

To find out how many feet a car traveling at 60 miles per hour travels in 10 seconds, we need to convert the speed from miles per hour to feet per second.

1 mile = 5,280 feet (5280 feet = 1 mile)

1 hour = 60 minutes

1 minute = 60 seconds

To convert 60 miles per hour to feet per second, we can use the following steps:

First, convert miles per hour to feet per minute:

60 miles/hour * 5280 feet/mile = 316,800 feet/hour

Then, convert feet per hour to feet per minute:

316,800 feet/hour / 60 minutes/hour = 5,280 feet/minute

Finally, convert feet per minute to feet per second:

5,280 feet/minute / 60 seconds/minute = 88 feet/second

Therefore, a car traveling at 60 miles per hour would travel 88 feet in 1 second. In 10 seconds, it would travel:

88 feet/second * 10 seconds = 880 feet.

Answer:

C. 33

Step-by-step explanation:

(√121) (√9) = (√11*11) (√3*3)

                  = (√11^2) (√3^2)

                  = (11)(3)

                   = 33

Step-by-step explanation:

triangle EFG is an isosceles triangle

angle G

= 180°-58°

= 122° (adj. angles on a str. line)

angle F

= (180°-122°)÷2

= 29° (angles in a triangle)