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choli [55]
3 years ago
8

Find the points on the cone z2 = x2 + y2 that are closest to the point (6, 2, 0).

Mathematics
1 answer:
aliya0001 [1]3 years ago
7 0

Answer:

The closest points on the cone are;

(6, 2, -√10) and (6, 2, √10)

Step-by-step explanation:

Let B(x, y, z) denote a point on the cone.

Therefore, the distance between the points (6, 2, 0) and B(x, y, z) is;

d = √[(x - 6)² + (y - 2)² + (z - 0)²]

d = √[(x - 6)² + (y - 2)² + z²]

Since we are given that z² = x² + y², we now have;

d = √[(x - 6)² + (y - 2)² + x² + y²]

Taking the square of both sides gives;

d² = [(x - 6)² + (y - 2)² + x² + y²]

x² is an increasing function. Thus, minimizing d is also the same as to minimize f (x, y) = d²

Thus, f' = 0. So;

df/dx = 2(x - 6) + 2x = 0

2x - 12 + 2x = 0

4x = 12

x = 12/4

x = 3

Similarly,

df/dy = 2(y - 2) + 2y = 0

2y - 4 + 2y = 0

4y - 4 = 0

4y = 4

y = 4/4

y = 1

Now,from earlier;

z² = x² + y²

Thus;

z = ±√(3² + 1²)

z = ±√10

Thus, the closest points on the cone are;

(6, 2, -√10) and (6, 2, √10)

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Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two da
KengaRu [80]

Answer:

12+17+y+z or 29+y+z.

Step-by-step explanation:

We have been given that Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two days. One expression for the total miles he will run is 12+y+17+z.

The Commutative Property of Addition states that we can add numbers in any order. For example a and b be two numbers.

According to Commutative Property of Addition a+b=b+a.

Similarly, we can write an equivalent expression to our given expression as:

12+17+y+z

We can simplify our expression as:

29+y+z.

Therefore, our required expression would be 12+17+y+z or 29+y+z.

4 0
3 years ago
Write the following inequality in slope-intercept form for -4 x + 2y > -8
Aleksandr-060686 [28]

Answer:

The answer is "y < 2x - 4".

Step-by-step explanation:

Write in <u>slope-intercept</u> form, the formula is "y = mx + b"

6 0
2 years ago
Write the cubic polynomial function f(x)in expanded form with zeros −6,−5,and −1, given that f(0)=60
RSB [31]

Answer:

f(x)=2x^{3}+24x^{2}+82x+60

Step-by-step explanation:

we know that

The roots of the polynomial are the values of x when the value of the polynomial f(x) is equal to zero

The roots of the polynomial function are

x=-6 -----> (x+6)=0

x=-5 -----> (x+5)=0

x=-1 -----> (x+1)=0

The equation of the cubic polynomial is

f(x)=a(x+6)(x+5)(x+1)

where

a is the leading coefficient

Remember that

f(0)=60

That means ------> For x=0 the value of f(x) is equal to 60

substitute the value of x and the value of y in the function and solve for a

60=a(0+6)(0+5)(0+1)

60=a(6)(5)(1)

60=30a

a=2

so

f(x)=2(x+6)(x+5)(x+1)

Applying the distributive property

Convert to expanded form                  

f(x)=2(x+6)(x+5)(x+1)\\\\f(x)=2(x+6)(x^{2}+x+5x+5)\\\\f(x)=2(x+6)(x^{2}+6x+5)\\\\f(x)=2(x^{3}+6x^{2}+5x+6x^{2}+36x+30)\\\\f(x)=2x^{3}+24x^{2}+82x+60

3 0
3 years ago
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IRINA_888 [86]

First we identify the percent's as decimals

1%=0.01                    64%=0.64

Then we multiply our decimals by 300

0.01 x 300 = 3          0.64 x 300 = 192

If you notice, 64 x 3 is equal to 192

64 x 3 = 192

So the answers are 3 and 192

(Hope this helps!)

8 0
2 years ago
Louisa’s restaurant bill comes to $75.50 and leave a 15% tip what is Louisa’s total restaurant bill?
Rufina [12.5K]

Answer: $86.83 is the total amount and $11.33 is the total of added tip

Step-by-step explanation:

3 0
3 years ago
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