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3 years ago

Use the identities A = A ∩ S and S = B ∪ B and a distributive law to prove that:

Mathematics

1 answer:

bonufazy [111]3 years ago

4 0

Answer:

Given that

A = A ∩ S -- (1)

S = B ∪ B -- (2)

To prove:

A = (A ∩ B) ∪ (A ∩ B)

(A ∩ B) ∪ (A ∩ B)

= [(A∪A) ∩ (A∪B)] ∩ [(B∪A) ∩ (B∪B)]

=[A ∩ (A∪B)] ∩ [(A∪B) ∩ S]

=A ∩ (A∪B)

Hence proved.

2) If B ⊂ A then A = B ∪ (A ∩ B)

R.H.S = B ∪ (A ∩ B)

= (B ∪ A) ∩ (B∪B) --(3)

As B is subset of A so

(B ∪ A) = A

From (2)

(B ∪ B) = S

(3) becomes

=A ∩ S

from (1)

A ∩ S = A

Hence proved

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A rancher wants to fence in an area of 1500000 square feet in a rectangular field and then divide it in half with a fence down t

VLD [36.1K]

Answer:

6000 ft

Step-by-step explanation:

Let length of rectangular field=x

Breadth of rectangular field=y

Area of rectangular field= $1500000$ square ft

Area of rectangular field= $l\times b$

Area of rectangular field= $x\time y$

$1500000=xy$

$y=\frac{1500000}{x}$

Fencing used ,P(x)= $x+x+y+y+y=2x+3y$

Substitute the value of y

P(x)= $2x+3(\frac{1500000}{x})$

$P(x)=2x+\frac{4500000}{x}$

Differentiate w.r.t x

$P'(x)=2-\frac{4500000}{x^2}$

Using formula: $\frac{dx^n}{dx}=nx^{n-1}$

$P'(x)=0$

$2-\frac{4500000}{x^2}=0$

$\frac{4500000}{x^2}=2$

$x^2=\frac{4500000}{2}=2250000$

$x=\sqrt{2250000}=1500$

It is always positive because length is always positive.

Again differentiate w.r.t x

$P''(x)=\frac{9000000}{x^3}$

Substitute x=1500

$P''(1500)=\frac{9000000}{(1500)^3}>0$

Hence, fencing is minimum at x=1 500

Substitute x=1 500

$y=\frac{1500000}{1500}=1000$

Length of rectangular field=1500 ft

Breadth of rectangular field=1000 ft

Substitute the values

Shortest length of fence used= $2(1500)+3(1000)=6000 ft$

Hence, the shortest length of fence that the rancher can used=6000 ft

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3 years ago

confirm or disprove ernest's work by selecting two different points and applying the slope formula. be sure to identify the poin

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Answer:

Part a) Ernest's calculation was correct

Part b) Ernest's calculation was correct

Step-by-step explanation:

The picture of the question in the attached figure

<u><em>The complete question is</em></u>

Ernest calculated the slope of the graph during the uphill climb to be 10. His work is shown below.

Points: (60.900). (15, 450)

Slope : 900-450/60-15=450/45=10

Part a) Determine whether his calculation was correct or incorrect, and then explain why.

Part b) Confirm or disprove Ernest's work by selecting two different points and applying the slope formula. be sure to identify the points you chose

Part a)

we know that

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

we have the points

A(60,900),B(15,450)

Remember that

$m_A_B=m_B_A$

step 1

Find $m_A_B$

substitute in the formula

$m_A_B=\frac{450-900}{15-60}\\\\m_A_B=\frac{-450}{-45}=10$

step 2

Find $m_B_A$

substitute in the formula

$m_B_A=\frac{900-450}{60-15}\\\\m_B_A=\frac{450}{45}=10$

therefore

Ernest's calculation was correct, because $m_A_B=m_B_A$

Part b) selecting two different points

we take the points

(0,300) and (50,800) -----> see the attached figure

substitute in the formula of slope

$m=\frac{800-300}{50-0}$

$m=\frac{500}{50}$

$m=10$

therefore

Ernest's calculation was correct

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2 years ago

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Answer:

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Step-by-step explanation:

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3 years ago

How to solve -2(-4f+5)=-7+2f-9

Damm [24]

Answer:

$f=\frac{3}{5}$

Step-by-step explanation:

$-2(4f+5)=-7+2f-9$

First, use PEMDAS to distribute the -2 to the parenthesis.

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Combine like terms on the right side.

$-8f - 10 = -16+2f$

Now, you may either subtract the f's or the constants. I'll subtract the constants first, meaning I will add 10 to both sides.

$-8f = -6 + 2f$

Now subtract the 2f from both sides.

$-10f = -6$

Divide both sides by 10.

$f = \frac{-6}{-10}$

Simplify.

$f = \frac{3}{5}$

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2 years ago

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