Two trains leave the station at the same time, one heading east and the other west. The eastbound train travels 20 miles per hour faster than the westbound train. If the two trains are 750 miles apart after 5 hours, what is the rate of the eastbound train?

Given two numbers x and y such that:

x + y = 12   ...    (1)

from  equation (1) 

y = 12 - x  

Using this value of y, we represent xy as

xy = f(x)= x(12 - x)

 f(x) = 12x - x^2

Differentiating the above function:

f'(x) = 12 - 2x

Maximum value of f(x) occurs at point for which f'(x) = 0.

Equating f'(x) to 0 we get:

12 - 2x = 0

 2x =  12

> x = 12/2 = 6

Substituting this value of x in equation (2):

y = 12 - 6 = 6

Therefore, value of xy is maximum when:

x = 6 and y = 6

The maximum value of xy = 6*6 = 36

The two numbers that sum up to 12 and maximize the product are 6 and 6.

Let's denote the two numbers by x and y. We know that x + y = 12, and we want to maximize the product P = xy.

First, express y in terms of x,

y = 12 - x

P = x(12 - x)
= 12x - x²

To find the maximum value of P, we can take the derivative of P with respect to x and set it to zero,

dP/dx = 12 - 2x

Set dP/dx to 0: 12 - 2x = 0

x = 6

Since x + y = 12, if x = 6, then y = 12 - 6 = 6.

The two numbers are 6 and 6.
This combination will maximize the product, which is 6 * 6 = 36.

In conclusion, the two numbers that add up to 12 and maximize their product are both 6.

1. Points M and N have coordinates (-4,0) and (4,2), respectively.

Then vector [tex]\overrightarrow{MN}=(4-(-4),2-0)=(8,2)[/tex] is perpendicular to the neede line.

2. Write the equation of line that passes through the point P(2,-4) and is perpendicular to vector  [tex]\overrightarrow{MN}=(8,2):[/tex]

[tex]8(x-2)+2(y+4)=0,\\8x-16+2y+8=0,\\8x+2y-8=0,\\4x+y-4=0.[/tex]

3. Find the point on x-axis, that lies on the perpendicular line.

When y=0, then 4x-4=0, x=1 and point (1,0) lies on perpendicular line.

Answer: correct choice is C.

Answer:  The required sales tax rate is 4.83%.

Step-by-step explanation:  Given that Percy paid 24.10 for a basketball and the price of a basketball was 22.99.

We are to find the sales tax rate.

According to the given information, the sales tax is given by

[tex]S.T.=24.10-22.99=1.11.[/tex]

Therefore, the sales tax rate is given by

[tex]\dfrac{1.11}{22.99}\times100\%=\dfrac{111}{22.99}\%=4.83\%.[/tex]

Thus, the required sales tax rate is 4.83%.

Answer:

Step-by-step explanation:

To fill the entire vase:

pi(3)^2(18)= 508.9 cubic centimeters

To fill vase 3/4 of the way:

3/4= .75

So then you take what it takes to fill the vase (508.9) and multiply that by .75 which equals 381.70 cubic centimeters.

3/4 of the volume of water needed to fill the vase is 162π cm^3.

To find how much water is needed to fill the vase 3/4 of the way, we first need to calculate the volume of the vase and then determine 3/4 of that volume.

The formula to calculate the volume of a cylinder is:

[tex]Volume = \pi * radius^2 * height[/tex]

Given:

Radius (r) = 3 cm

Height (h) = 18 cm

Volume of the entire vase:

[tex]Volume = \pi * (3 cm)^2 * 18 cm\\Volume = \pi* 9 cm^2 * 18 cm\\Volume = 162\pi cm^3[/tex]

Now, to find 3/4 of the volume, multiply the total volume by 3/4:

[tex]3/4 * 162\pi cm^3 = 3 * 54\pi cm^3 = 162\pi cm^3[/tex]

So, 3/4 of the volume of water needed to fill the vase is 162π cm^3.

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f + 0.2 =-3

f = -3 -0.2

f = -3.2

divide 125 by 5

125 /5 = 25 meters per minute

Answer:

The rate per min is, 25 meters/minute

Step-by-step explanation:

Unit rate is defined as when rates are expressed as a quantity of 1, such as 4 feet per second or 6 miles per hour, they are called unit rates.

Given the statement: a machine laying underground cable can place 125 meters of cable in 5 minutes.

Then by definition:

rate per minute  = [tex]\frac{125}{5} = 25[/tex] meters /minute

therefore, the rate per minute is, 25 meters/ minute

Final answer:

The speed of the slower plane is 425 mph, and the speed of the faster plane is 460 mph, determined by using the concept of relative speed and algebra to solve the given equation.

Explanation:

To solve the problem of two planes flying towards each other, we need to use the concept of relative speed. We know that the two planes are 3540 miles apart and that they pass each other after 4 hours. The speeds of the planes differ by 35 mph. The relative speed of the two planes combined is the distance divided by the time, so we calculate it as follows:

Relative Speed = Total Distance / Time = 3540 miles / 4 hours = 885 mph

So, if we denote the speed of the slower plane as S mph, the speed of the faster plane will be S + 35 mph. Since their combined speed is the relative speed, we can set up the following equation:

S + (S + 35) = 885

From this equation, we need to find the value of S which represents the speed of the slower plane. We can then add 35 to S to find the speed of the faster plane. Here's how it's done step by step:

Therefore, the speed of the slower plane is 425 mph and the speed of the faster plane is 460 mph.

The tangent line is the point that touches a graph at a point.

The value of x at the tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]

The function is given as:

[tex]\mathbf{y=-9x^2 - 2x}[/tex]

Differentiate both sides with respect to x

[tex]\mathbf{y' =-18x - 2}[/tex]

Set the above equation to 0, to calculate the value of x

[tex]\mathbf{-18x - 2 = 0}[/tex]

Collect like terms

[tex]\mathbf{-18x = 2 }[/tex]

Divide both sides by -18

[tex]\mathbf{x = -\frac 19 }[/tex]

Hence, the value of x when at tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]

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A rectangular garden must have an area of 64 square feet then the minimum perimeter of the garden is 32 feet and this can be determined by using the formula of the perimeter of a rectangle.

Given :

A rectangular garden must have an area of 64 square feet.

The area of a rectangle is given by:

[tex]\rm A =l\times w[/tex]

where 'l' is the length of the rectangle and 'w' is the width of the rectangle.

Given that area of the rectangular garden is 64 square feet that is:

64 = lw

[tex]\rm w = \dfrac{64}{l}[/tex]   ---- (1)

Now the perimeter of a rectangle is given by:

P = 2(l + w)

Put the value of w in the above equation.

[tex]\rm P = 2 (l + \dfrac{64}{l})[/tex]   ---- (2)

For minimum perimeter differentiate the above equation with respect to the length of the garden.

[tex]\rm P' = 2 - \dfrac{128}{l^2}[/tex]  

Now, equate the above equation to zero.

[tex]\rm 0 = 2-\dfrac{128}{l^2}[/tex]

[tex]l^2 = 64[/tex]

[tex]l = 8[/tex]

Now put the value of l in equation (2).

[tex]\rm P = 2(8 + \dfrac{64}{8})[/tex]

P = 32 feet.

A rectangular garden must have an area of 64 square feet then the minimum perimeter of the garden is 32 feet.

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We are given the equations:

5 x + 11 y = 49                    --> eqtn 1

u = 3 x^2 + 6 y^2               --> eqtn 2

Rewrite eqtn 1 explicit to y:

11 y = 49 – 5 x

y = (49 – 5x) / 11               --> eqtn 3

Substitute eqtn 3 to eqtn 2:

u = 3 x^2 + 6 [(49 – 5x) / 11]^2

u = 3 x^2 + 6 [(2401 – 490 x + 25 x^2) / 121]

u = 3 x^2 + 14406/121 – 2940x/121 + 150x^2/121

u = 4.24 x^2 – 24.3 x + 119.06

Derive then set du/dx = 0 to get the maxima:

du/dx = 8.48 x – 24.3 = 0

solving for x:

8.48 x = 24.3

x = 2.87

so y is:

y = (49 – 5x) / 11 = (49 – 5*2.87) / 11

y = 3.15

Answer:

George will choose some of each commodity but more y than x.

At least 1, possibly as many as 3.

There are 6 pairs of integers among those from -1 to -12 that will sum to -13. If you choose 7 integers, you may only choose one pair, or you may choose as many as three pairs.

One to three pairs will sum to -13.

Answer:

TQ =  11.4

Step-by-step explanation:

Given : In rectangle PQRS, PQ = 18, PS = 14, and PR = 22.8. Diagonals PR and QS intersect at point T.

To find : What is the length of TQ .

Solution : We have given rectangle PQRS

Side PQ = 18.

Side PS = 14.

Diagonal  PR = 22.8 .

Properties of rectangle : (1)  Opposite sides of rectangle are equals.

(2) Diagonals of rectangle are equal .

(3)  Diagonals of rectangle bisect each other.

Then by second property :

Diagonal PR= QS .

QS = 22.8

By the Third property  TQ = [tex]\frac{1}{2} * QS[/tex].

TQ =  [tex]\frac{1}{2} * 22.8 [/tex].

TQ =  11.4

Therefore, TQ =  11.4

The probability that the first head comes up in an odd number of tosses can be determined using geometric probability.

When tossing a coin repeatedly until we get a head, the probability that the first head comes up in an odd number of tosses can be determined using geometric probability. Since the probability of getting a head in one toss is 0.5, the probability of getting a head in an odd number of tosses (one, three, five, etc.) can be calculated using the formula:

P(odd number of tosses) = P(tail) * P(tail) * P(tail) * ... * P(head)

The number of terms in the product is determined by the number of tosses required to get the first head. For example, if it takes 3 tosses to get the first head, the formula becomes:

P(odd number of tosses) = P(tail) * P(tail) * P(head)

Since the probability of getting a tail is 0.5 and the probability of getting a head is also 0.5, the formula simplifies to:

P(odd number of tosses) = 0.5 * 0.5 * 0.5 * ... * 0.5 = (0.5)^n

Where 'n' is the number of tosses required to get the first head.

Answer:

The monthly payment is $364.42

B is correct

Step-by-step explanation:

The monthly payment on $13,300 financed at 7.9 percent for 4 years.

If the monthly payment per $100 is $2.74

Financed amount = $13,300

We are given monthly payment for $100 is $2.74

It means we pay $2.74 for $100.

Now we find how many number of $100 in $13,300

Number of $100 in $13,300 [tex]=\dfrac{13300}{100} = 133[/tex]

133 number of hundred in $13,300

For each $100 monthly payment  = $2.74

                           For 133 payment = 133 x 2.74

Monthly Payment for $13,300 finance = $364.42

Hence, The monthly payment is $364.42

Final answer:

To find the greatest common divisor of 273 and 3019 using Euclid's algorithm, we divide the larger number by the smaller and use the remainder to repeat the process until we reach a remainder of 0. The gcd of 273 and 3019 is determined to be 1, indicating that they are coprime.

Explanation:

Using Euclid's Algorithm to Find the GCD

Euclid's algorithm is a method to determine the greatest common divisor (gcd) of two numbers. To find the gcd of 273 and 3019, we perform the Euclidean division repeatedly until we get a remainder of zero. The last non-zero remainder will be the gcd of the given numbers.

Divide 3019 by 273 to get a quotient of 11 and a remainder of 46.

Next, divide 273 by 46 to get a quotient of 5 and a remainder of 43.

Then, divide 46 by 43 to get a quotient of 1 and a remainder of 3.

Finally, divide 43 by 3 to get a quotient of 14 and a remainder of 1.

Now, divide 3 by 1 to get a quotient of 3 and a remainder of 0.

Since the last non-zero remainder is 1, the greatest common divisor (gcd) or g of 273 and 3019 is 1. Thus, 273 and 3019 are coprime or relatively prime to each other.

20% = 0.20

 multiply original price ( $0) by 0.20 for the amount of discount then subtract it

80*0.20 = 16.00

 she will save $16

80-16 = 64

 she will pay $64 after the discount