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Ganezh [65]
3 years ago
5

Which points are on the graph of the equation -3x + 6y + 5 = -7?

Mathematics
1 answer:
LiRa [457]3 years ago
3 0

Answer:

(-4,-4) (0,-2) (4,0) (2,8)

Step-by-step explanation:

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Find the point on the parabola y^2 = 4x that is closest to the point (2, 8).
guapka [62]

Answer:

(4, 4)

Step-by-step explanation:

There are a couple of ways to go at this:

  1. Write an expression for the distance from a point on the parabola to the given point, then differentiate that and set the derivative to zero.
  2. Find the equation of a normal line to the parabola that goes through the given point.

1. The distance formula tells us for some point (x, y) on the parabola, the distance d satisfies ...

... d² = (x -2)² +(y -8)² . . . . . . . the y in this equation is a function of x

Differentiating with respect to x and setting dd/dx=0, we have ...

... 2d(dd/dx) = 0 = 2(x -2) +2(y -8)(dy/dx)

We can factor 2 from this to get

... 0 = x -2 +(y -8)(dy/dx)

Differentiating the parabola's equation, we find ...

... 2y(dy/dx) = 4

... dy/dx = 2/y

Substituting for x (=y²/4) and dy/dx into our derivative equation above, we get

... 0 = y²/4 -2 +(y -8)(2/y) = y²/4 -16/y

... 64 = y³ . . . . . . multiply by 4y, add 64

... 4 = y . . . . . . . . cube root

... y²/4 = 16/4 = x = 4

_____

2. The derivative above tells us the slope at point (x, y) on the parabola is ...

... dy/dx = 2/y

Then the slope of the normal line at that point is ...

... -1/(dy/dx) = -y/2

The normal line through the point (2, 8) will have equation (in point-slope form) ...

... y - 8 = (-y/2)(x -2)

Substituting for x using the equation of the parabola, we get

... y - 8 = (-y/2)(y²/4 -2)

Multiplying by 8 gives ...

... 8y -64 = -y³ +8y

... y³ = 64 . . . . subtract 8y, multiply by -1

... y = 4 . . . . . . cube root

... x = y²/4 = 4

The point on the parabola that is closest to the point (2, 8) is (4, 4).

4 0
3 years ago
Al saves pennies. He agreed to give 3/7 of his pennies to Bev if she would give 3/7 of what she got from Al to Carl and if Carl
ser-zykov [4K]

Answer:

2744 pennies.

Step-by-step explanation:

Let x represent the number of pennies Al have initially.

We have been given that Al agreed to give 3/7 of his pennies to Bev, so number of pennies Bev will get would be \frac{3}{7}x.

We are also told that Bev gave 3/7 of what she got from Al to Carl, so number of pennies that Carl will get would be \frac{3}{7}x\cdot \frac{3}{7}=\frac{9}{49}x.

Further, Carl gave 3/7 of what he got from Bev to Dani, so number of pennies that Dani will get would be \frac{9}{49}x\cdot \frac{3}{7}=\frac{27}{343}x.

We are also told that Dani received 216 pennies. Now, we will equate 216 to \frac{27}{343}x and solve for x as shown below:

\frac{27x}{343}=216

\frac{27x}{343}*343=216*343

27x=74088

\frac{27x}{27}=\frac{74088}{27}

x=2744

Therefore, Al have 2744 pennies initially.

3 0
3 years ago
An animal shelter spends $3.00 per day to care for each cat and $6.50 per day to care for each dog. Amelia noticed that the shel
Ede4ka [16]

Answer:

There were 15 cats at the shelter on Friday.

Step-by-step explanation:

Cost of caring each cat = $3.00

Cost for caring each dog = $6.50

Total money spent by shelter = $84.00

Total Cats and Dogs in the shelter = 21

Let the number of cats in the shelter = k

So, the number of dogs in shelter =  21 -k

Also according to the question:

Total expenditure on cats + total expenditure on dogs = $84.00

or, k ($3.00) + (21-k)($6.50)  = $84.00

or, 3k + 136.5 - 6.5 k = 84

or, -3.5 k = 84 - 136.5 = -52.5

⇒3.5 k = 52.5  or k = 52.5/3.5 = 15

⇒  k = 15

Hence, the number of cats in the shelter was 15 on Friday.

7 0
3 years ago
motel clerk counts his $1 and $10 bills at the end of a day. He finds that he has a total of 57 bills having a combined monetary
madreJ [45]
You can use a matrix equation to discover that the motel clerk has 45 $1 bills and 12 $10 bills.
4 0
3 years ago
An angle measures 47.2° less than the measure of its supplementary angle. What is the measure of each angle? on ixl
SVEN [57.7K]

Answer:

The measure of one angle is 113.6^o, and the measure of the other one is 66.4^o

Step-by-step explanation:

Recall that supplementary angles are those whose addition renders 180^o

We need to find the measure of two such angles whose difference is precisely 47.2^o.

Let's call such angles x and y, and consider that angle x is larger than angle y, so we can setup the following system of equations:

x+y=180^o\\x-y=47.2^o

We can now solve this by simply combining term by term both equations, thus  cancelling the term in "y", and solving first for "x":

x+y=180^o\\x-y=47.2^o\\\\2x=180^o+47.2^o\\2x=227.2^o\\x=113.6^o

So, now we have the answer for one of the angles (x), and can use either equation from the system to find the measure of angle "y":

y=180^o-x\\y=180^o-113.6^o\\y=66.4^o

5 0
3 years ago
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