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2 years ago

Find a point on the curve x^3+y^3=11xy other than the origin at which the tangent line is horizontal.

Mathematics

1 answer:

attashe74 [19]2 years ago

3 0

Compute the derivative dy/dx using the power, product, and chain rules. Given

x³ + y³ = 11xy

differentiate both sides with respect to x to get

3x² + 3y² dy/dx = 11y + 11x dy/dx

Solve for dy/dx :

(3y² - 11x) dy/dx = 11y - 3x²

dy/dx = (11y - 3x²)/(3y² - 11x)

The tangent line to the curve is horizontal when the slope dy/dx = 0; this happens when

11y - 3x² = 0

y = 3/11 x²

(provided that 3y² - 11x ≠ 0)

Substitute y into into the original equation:

x³ + (3/11 x²)³ = 11x (3/11 x²)

x³ + (3/11)³ x⁶ = 3x³

(3/11)³ x⁶ - 2x³ = 0

x³ ((3/11)³ x³ - 2) = 0

One (actually three) of the solutions is x = 0, which corresponds to the origin (0,0). This leaves us with

(3/11)³ x³ - 2 = 0

(3/11 x)³ - 2 = 0

(3/11 x)³ = 2

3/11 x = ³√2

x = (11•³√2)/3

Solving for y gives

y = 3/11 x²

y = 3/11 ((11•³√2)/3)²

y = (11•³√4)/3

So the only other point where the tangent line is horizontal is ((11•³√2)/3, (11•³√4)/3).

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Team tool bella canoed 15 3/4 miles in 5/4 hours. What was their average rate of speed in mph

luda_lava [24]

Average speed is 12.6 miles per hour

<em><u>Solution:</u></em>

Given that team tool bella canoed $15\frac{3}{4}$ miles in $\frac{5}{4}$ hours

To find: average rate of speed in miles per hour

From given information,

$distance = 15\frac{3}{4} \text{ miles } = \frac{15 \times 4+3}{4} = \frac{63}{4} \text{ miles}$

$time = \frac{5}{4} \text{ hours}$

To find the speed we use the below formula,

$speed = \frac{distance}{time}$

Substituting the given values we get,

$speed = \frac{\frac{63}{4}}{\frac{5}{4}}\\\\speed = \frac{63}{4} \times \frac{4}{5} = \frac{63}{5} = 12.6$

Thus average speed is 12.6 miles per hour

7 0

2 years ago

A veterinary researcher takes a random sample of 60 horses presenting with colic. The average age of the random sample of horses

Licemer1 [7]

Answer:

Probability that a sample mean is 12 or larger for a sample from the horse population is 0.0262.

Step-by-step explanation:

We are given that a veterinary researcher takes a random sample of 60 horses presenting with colic. The average age of the random sample of horses with colic is 12 years. The average age of all horses seen at the veterinary clinic was determined to be 10 years. The researcher also determined that the standard deviation of all horses coming to the veterinary clinic is 8 years.

So, firstly according to Central limit theorem the z score probability distribution for sample means is given by;

Z = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\bar X$ = average age of the random sample of horses with colic = 12 yrs

$\mu$ = average age of all horses seen at the veterinary clinic = 10 yrs

$\sigma$ = standard deviation of all horses coming to the veterinary clinic = 8 yrs

n = sample of horses = 60

So, probability that a sample mean is 12 or larger for a sample from the horse population is given by = P( $\bar X$ $\geq$ 12)

P( $\bar X$ $\geq$ 12) = P( $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ $\geq$ $\frac{12-10}{\frac{8}{\sqrt{60} } }$ ) = P(Z $\geq$ 1.94) = 1 - P(Z < 1.94)

= 1 - 0.97381 = 0.0262

Therefore, probability that a sample mean is 12 or larger for a sample from the horse population is 0.0262.

4 0

2 years ago

Classify each number below as an integer or not.

fomenos

Answer:

1. yes

2. no

3. yes

4. no

5. yes

Step-by-step explanation:

The method i use is if the number is not a fraction than it is an integer. besides 6 for some reason.

Here is the catch if the fraction can be simplified into a whole number than it is an integer.

6 0

3 years ago

Which one is it A. B. C. or D.

san4es73 [151]

The answer will be c
Hope it's help you

6 0

2 years ago

A flag staff is fixed at one corner of a rectangular room of side 20×10m.The angle of elevation on the top of the flag from oppo

Vedmedyk [2.9K]

Answer:

The height is 12.9m

Step-by-step explanation:

First we have to find the distance from the corner of the flag to the opposite corner, for this we will use Pythagoras

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides

h² = s1² + s2²

h² = 10² + 20²

h² = 100 + 400

h² = 500

h = √500

h = 22.36

Now that we know this measurement we can calculate the height of the flagpole

well to start we have to know the relationship between angles, legs and the hypotenuse

α = 30

a: adjacent = 22.36

o: opposite = ?

h: hypotenuse

sin α = o/h

cos α= a/h

tan α = o/a

we see that it has (angle, adjacent, opposite)

is the tangent

tan α = o/a

tan 30 = o/22.36

tan30 * 22.36 = o

12.9 = o

The height is 12.9m

7 0

3 years ago