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a light is suspended at a height h above the floor. the illumination, i, at a point p is inversely proportional to the square of the distance, r, from the point p to the light and directly proportional to the cosine of the angle theta , I = k cos x / r² can be used to find the illumination at point p.

Light must be at the height of 7.07 for maximum illumination.

I = k cos x / r²

r² = h² + 10²

cos θ = h / [tex]\sqrt{h^{2} +10^{2} }[/tex]

then

I =  [tex]k \frac{h}{(h^{2} + 100)^{2/3}}[/tex]

on differentiating, we get,

[tex]\frac{dI}{dh} = k\frac{h^{2} + 100^{3/2 - 3h^{2}(h^{2} + 100 )^{1/2} } }{(h^{2} +100)^{3} }[/tex]

[tex]= k\frac{100 - 2h^{2} }{(h^{2} + 100)^{5/2} } \\\frac{dI}{dh} = 0\\ It gives that \\h = \sqrt{50} = 7.07\\on differentiating again,\\\frac{d^{2} I}{dh^{2} } = k \frac{(6h^{3} - 900h )}{(h^{2} + 100)^{7/2} }[/tex]

for h = 7.07,

[tex]\frac{dI^{2} }{dh^{2} } = -10267.6[/tex]

So , we can conclude that the light must be at the height of 7.07 for maximum illumination.

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The remainder theorem conclude given that F(-5)=17

When, x= -5

x+5=0

So, F(-5) = 17

According to the remainder theorem, the result of dividing p(x) by (x - a) is p. (a). According to the factor theorem, (x - a) can only be a factor of p(x) if f(a) = 0. It is used to locate the rest. A linear polynomial's status as a factor of the input polynomial is determined using this method.

The remainder theorem is used to calculate the remainder when a polynomial is split by a linear polynomial. We are aware that long division of polynomials can be used to find the remainder after dividing one polynomial by another. The remainder can instead be discovered using the remainder theorem.

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Answer: 2 I think

Step-by-step explanation:

Answer:

volume of cylinder is πr²h so whatever your radius is then square it multiply by height. don't multiply by pi as it gives it to you in the answer

Answer:

- 8 < x < - 4

Step-by-step explanation:

- 68 < 8x - 4 < - 36 ( add 4 to each interval )

- 64 < 8x < - 32 ( divide each interval by 8 )

- 8 < x < - 4

The algebraic expression that models the situation when Jenny had $105, but she is spending $13 per week is 105 - 13w.

Let the number of weeks be represented by x

Therefore, since Jenny had $105, but she is spending $13 per week, the appropriate expression will be:

= 105 - (13 × w)

= 105 - 13w

The expression is 105 - 13w.

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Answer:14% of the seats are empty

Step-by-step explanation:

7/50 = ?/100

50×2=100

so 7×2=?

?=14

14/100=14%

Answer:

es 25

Step-by-step explanation:

Answer:

b) 14

Step-by-step explanation:

Given problem,

→ -3 + |-17|

Let's solve the problem,

→ -3 + |-17|

→ -3 + 17

→ 14

Hence, the answer is 14.

From the given integer numbers, the states that have temperatures within 10ºC of Delaware are: California, Hawaii and Idaho.

The temperature in Delaware is of 5 ºC, hence for our range, we have that:

Alaska(< -5), Tennessee(>15) and WV(<-5) do not have temperatures in this range, hence the states that have temperatures within 10ºC of Delaware are: California, Hawaii and Idaho.

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Answer: True, common sense dictates this.

Answer: true

Step-by-step explanation:

Answer:

I think the answer is

Statement A

Step-by-step explanation:

Answer: nice for her

Step-by-step explanation:

Question:

You have $2.75. You want to buy trading cards that cost $0.25 each.

How many can you buy?

Answer:

11 trading cards

Step-by-step explanation:

If you do 2.75 ÷ 0.25 you would get 11 and to check the answer you could simply multiply you would do 0.25 x 11 to get the answer 2.75.

I hope it helps you!

~XxBells is a cute girlxX~

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Answer: y= 60x, y=50x

Step-by-step explanation:

Please Provide the whole question

The temperature in degrees celcius in which bacteria cannot survive is greater than 3.66

This is a way of representing a variable with another

Given the inequality below;

q/5 C + 32 ≥ 120

Substitute C = 120oF

q/5(120) + 32  ≥ 120

24q + 32 ≥ 120

24q  ≥ 120 - 32

24q  ≥ 88

q ≥ 88/24

q ≥ 3.66

Hence the temperature in degrees celcius in which bacteria cannot survive is greater than 3.66

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[tex]110\cdot 10^{4x}=55\implies 10^{4x}=\cfrac{55}{110}\implies 10^{4x}=\cfrac{1}{2}\implies 10^{4x}=2^{-1} \\\\\\ \log\left( 10^{4x} \right)=\log\left( 2^{-1} \right)\implies 4x\log_{10}\left( 10 \right)=-1\log_{10}(2)\implies 4x(1)=-\log(2) \\\\\\ 4x=-\log(2)\implies x=-\cfrac{\log(2)}{4}\implies {\LARGE \begin{array}{llll} x\approx -0.08 \end{array}}[/tex]

The resultant of the parallel force passes, x=2.8888 in and y=-5.4444 in

Taking moment about 'y'

∈Mx=RXx

x=260/90

x=2.8888 in

Taking moment about 'x'

∈My=RXy

y= -490/90

y= -5.4444 in

When two forces of equal strength act in the same direction within the same plane, with the counterforce in the centre, this is known as a parallel force system.

A see-saw is an illustration of this. The fulcrum in the middle provides the counterforce to keep the seesaw in a neutral position while the children apply the two forces at the ends. The significant vertical forces on an aeroplane while it is in flight is another illustration (see image at right).

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For the conditional statement, we have that:

The conditional statement has the following format:

p -> q.

For this problem, we have that the events are given as follows:

Hence p -> q is a true statement, as the set of whole numbers are the naturals plus the 0.

The inverse is given by:

~p -> ~q.

In which ~ represents the negation symbol.

Hence:

If a number is not a natural number, then the number is also not a whole number.

This statement has a false truth value, as the number 0 is whole but not natural.

The converse is given by:

q -> p.

Hence:

If a number is a whole number, then it is also a natural number.

This statement has a false truth value, as the number 0 is whole but not natural.

The contra-positive is given by:

~q -> ~p.

Hence:

If a number is not a whole number, then it is also not a natural number.

This statement has a true truth value, as all natural numbers are whole numbers.

The bi-conditional is given by:

p <-> q

In which <-> represents the term if, and only if.

Hence:

A number is a natural number if, and only if, it is also a whole number.

This statement has a true truth value, as all natural numbers are whole numbers.

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The zeroes of the polynomial functions are listed below:

Polynomials are algebraic constructions whose form is summarized by the following sum:

p(x) = ∑ cₙ · xⁿ, for n ∈ {0, 1, ..., m}

An alternative form to describe polynomials is using products of binomials of the form (x - r), where r is a root (zero) of the polynomial. The root of the polynomial is a value such that the result of the expression is equal to zero.

Then, we proceed to find the zeros of each polynomial function:

1) f(x) = 2 · x · (3 - x) · (x + 1). Roots: 2, 3, - 1.

2) f(x) = 2 · (x - 3) · (x + 2)² · (x - 1). Roots: 3, - 2 (multiplicity 2), 1

3) f(x) = x³ · (x - 2) · (x + 3). Roots: 0 (multiplicity 3), 2, - 3.

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Answer:

Min = 0

Max = 51

M = 25.5

Q1 = 16.5

Q3 = 39

Hope this helps! <3

According to the question, to calculate break-even quantity with the given data. And the data is in dollars which is as follows:

Fixed cost (f(c)) = [tex]$1400[/tex] per week;

Variable cost (v(c)) = [tex]$2[/tex] per unit

Revenue = [tex]$6[/tex] per unit

Let us consider the break-even quantity to be ‘a’.

As per break-even point formula: Total costs = Total Revenue

And Total costs = Fixed cost + variable cost ;

      Total Revenue = (price per unit)(quantity sold)

Using above formulas, the above expression can be re-written as:

Fixed cost + variable cost = (price per unit)(quantity sold)

[tex]($1400)+($2a) = ($6a)[/tex]

[tex]4a = 1400[/tex]

[tex]a = 350[/tex]

Therefore, the break-even quantity is [tex]350[/tex] units.

What is the break-even quantity?

Break-even quantity in the firm is the incremental value which is used to sell in order to cover the cost of the marketing program or other companies investments. And in this, the total cost is equal to the total revenue.

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The probability that the four houses selected for a free newspaper will all be even numbered houses is: 1/42.

Probability refers to the possibility of the occurrence of an event.

P(E) = probability of occurrence of an event E = Number of favorable outcomes, divided by total number of outcomes

Now, according to question,

There are 10 houses numbered 1-10. Thus the number of even numbered houses = 5 (2, 4, 6, 8, 10)

The probability that the first selected house for free newspapers is even numbered = 5/10 = 1/2

The probability that the second selected house for free newspapers is even numbered = 4/9

The probability that the third selected house for free newspapers is even numbered = 3/8

The probability that the fourth selected house for free newspapers is even numbered = 2/7

The final probability that the four houses selected for a free newspaper will all be even numbered houses is = 1/2(4/9)(3/8)(2/7) = 1/42

Hence, The probability that the four houses selected for a free newspaper will all be even numbered houses is: 1/42.

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Wayne's actual act score is 18

A z-score tells you where the score lies on a normal distribution curve.

A standard deviation is a measure of how dispersed the data is in relation to the mean.

Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.

Given,

The mean = 21

Standard deviation =5

z-score = -0.6

We know z-score = (x-mean)/ Standard deviation

Substitute the values

-0.6 = [tex]\frac{x-21}{5}[/tex]

x-21 = -0.6 ×5

x-21 = -3

x=-3+21

x=18

Hence, the Wayne's actual act score is 18

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Answer:

False

Step-by-step explanation:

Without a graph, figure, or any other context the cost of grapefruit in 1995 in not related to the other years by any other facts or figures.

Answer:

1. x = 2/3y + 8/3

2. x=−y+1

Step-by-step explanation:

Solve for x

3x−2y=8

Add 2y to both sides.

3x−2y+2y = 8+2y

3x=2y+8

Divide both sides by 3.

3x/3 = 2y+8/3

x = 2/3y + 8/3

Answer:

x = 2/3y + 8/3

#2

solve for x.

x+y=1

Add -y to both sides.

x+y+−y=1+−y

x=−y+1

Answer:

x=−y+1

solve for y.

x+y=1

Add -x to both sides.

x+y+−x=1+−x

y=−x+1

Answer:

y=−x+1

Sorry if this is too much

Subtracting the maximum sodium intake from the intake of the cereal bar, you can eat amounts of sodium less than 2,175 mg of sodium the rest of the day.

From the text, the intake has to be less than 2300 mg. From the table, each cereal bar has 125 mg of sodium.

You eat one cereal bar, hence the remaining amount you can eat is found as follows:

2300 - 125 = 2,175.

You can eat amounts of sodium less than 2,175 mg of sodium the rest of the day.

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