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8090 [49]
3 years ago
11

Explain how you could find the shortest distance from A(6, 5) to the line y = 5x – 10. (Use a diagram, be specific, and list all

your steps.
Mathematics
1 answer:
HACTEHA [7]3 years ago
3 0

Step-by-step explanation:

I cannot draw a diagram here.

but I can explain what to do in general.

the shortest distance from a point to a line is always via a connecting line that is perpendicular to the given line and his through the given point.

and then the distance from the given point to the intersection point is calculated.

every line is defined in the form like

y = ax + b

where a is the slope of the line, and b is the intersection point on the y-axis (the offset from point 0).

the slope of a line is the ratio y/x defining how many units y changes when x changes a certain amount of units.

in our example,

y = 5x - 10

5 (or rather 5/1) is the slope of the line.

it means that y grows by 5 units every time x grows by 1 unit.

a perpendicular line (cuts the original line with a 90 degree angle) has a related slope : it reverts x and y and flips the sign :

5/1 turns into -1/5

that means at the perpendicular line whenever x grows by 5 units, y goes down by 1 unit.

so, the first approach for the perpendicular line is

y = -1/5 x + b

to get b we use the given point (6, 5) that has to be in the perpendicular line.

5 = -1/5 × 6 + b

25/5 = -6/5 + b

31/5 = b

=> y = -1/5 x + 31/5

the intersecting point is now where both lines are equal

5x - 10 = -1/5 x + 31/5

25x - 50 = -x + 31

26x = 81

x = 81/26

y = 5×(81/26) - 10 = 405/26 - 260/26 = 145/26

the distance of the given point (6, 5) to the line intersection point (81/26, 145/26) is the calculated as

distance² = (6 - 81/26)² + (5 - 145/26)²

distance = sqrt((6-81/26)² + (5-145/26)²)

since the result was not requested here, I save us the calculation.

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A gardener wishes to design a rectangular rose garden with one side of the garden along a main road. The fencing for the 3 non-r
Reptile [31]

Answer:

There is no minimum cost for the fencing that can satisfy these conditions.

as the equation obtained for this problem is y = \frac{7x}{x-6}.

Step-by-step explanation:

let length of the rectangular rose garden be x

and breath of the rectangular rose garden be y

then the area of the garden = xy

cost of fencing a foot for the 3 non road sides = $6

and cost of fencing a foot for a road side = $8

total cost of fencing = 6x + 8x + 6y + 6y

so, total cost = 14x + 12y

cost of fencing for every square foot of the area =$2

So, 2xy = 14x + 12y

=> 2xy = 14x + 12y

Also, x and y has to satisfy x>0 and y>0

You can solve for y (or x if you prefer)

2xy-12y=14x

xy - 6y = 7x

y(x-6)=7x

y = \frac{7x}{x-6}

we can check that this function has no minimum value.

as we increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the product xy will increase.

So, there is not a minimum cost for fencing that can satisfy these conditions.

8 0
3 years ago
Read 2 more answers
The measures of angle ABC are given by the expression on the table (shown below)
amid [387]

Answer:

<A = ( 4 * 32 - 13 ) = 115

<C = ( x + 18 ) = 50

Step-by-step explanation:

(4x - 13 ) + 15 + ( x + 18 ) = 180 ( because the total angle of a triangle is 180)

5x + 20 = 180

5x = 160

x = 32

8 0
3 years ago
The sum of two numbers is odd. Can the quotient of the two numbers be an odd number?
Natalka [10]

Answer: No.

Step-by-step explanation:

I guess that here we have the statement:

If the sum of two numbers is odd----> can their quotient be an odd number?

first, for n an integer number, we have that:

an odd number can be written as 2n + 1

an even number can be written as 2n.

The sum of two numbers is only odd if one of them is odd and the other even.

Then we have a number that is 2n and other that is 2k + 1, for n and k integer numbers.

Now, let's see if the quotient can also be an odd number.

One way to think this is:

There is an odd number such that when we multiply it by another odd number, the result is an even number?

no, and i can prove it as:

let 2k + 1 be an odd number, and 2j + 1 other.

the product is:

(2k + 1)*(2j + 1) = 2*(2*k*j + k + j) + 1

and as k and j are integers, also does 2*k*j + k + j, so:

2*(2*k*j + k + j) + 1 is an odd number.

This says that the product of two odd numbers is always odd, then we never can have that the quotient between an even number and an odd number is odd.

4 0
3 years ago
Solve 2(x + 4) + 8x = 14 ​
Alexandra [31]
Use photo math. or mathpapa to help you solve the equation
7 0
2 years ago
What’s the probability?
son4ous [18]

Answer:

\frac{1}{8}

Step-by-step explanation:

The probability for a flip on either side of the coin is always 50% or \frac{1}{2}

So to get the probability for the three flips just do:

(\frac{1}{2}) ^{3}  or \frac{1}{2} × \frac{1}{2} × \frac{1}{2}

Either way you solve it the answer is \frac{1}{8}

6 0
2 years ago
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