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Ivahew [28]
3 years ago
5

For which triangle is the length of the hypotenuse an INTEGER?

Mathematics
1 answer:
artcher [175]3 years ago
4 0

A right triangle :) to find the length of a missing side, you can use the pythagorean theorem (a^2 + b^2 = c^2)

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Suppose Upper F Superscript prime Baseline left-parenthesis x right-parenthesis equals 3 x Superscript 2 Baseline plus 7 and Upp
Sedaia [141]

It looks like you're given

<em>F'(x)</em> = 3<em>x</em>² + 7

and

<em>F</em> (0) = 5

and you're asked to find <em>F(b)</em> for the values of <em>b</em> in the list {0, 0.1, 0.2, 0.5, 2.0}.

The first is done for you, <em>F</em> (0) = 5.

For the remaining <em>b</em>, you can solve for <em>F(x)</em> exactly by using the fundamental theorem of calculus:

F(x)=F(0)+\displaystyle\int_0^x F'(t)\,\mathrm dt

F(x)=5+\displaystyle\int_0^x(3t^2+7)\,\mathrm dt

F(x)=5+(t^3+7t)\bigg|_0^x

F(x)=5+x^3+7x

Then <em>F</em> (0.1) = 5.701, <em>F</em> (0.2) = 6.408, <em>F</em> (0.5) = 8.625, and <em>F</em> (2.0) = 27.

On the other hand, if you're expected to <em>approximate</em> <em>F</em> at the given <em>b</em>, you can use the linear approximation to <em>F(x)</em> around <em>x</em> = 0, which is

<em>F(x)</em> ≈ <em>L(x)</em> = <em>F</em> (0) + <em>F'</em> (0) (<em>x</em> - 0) = 5 + 7<em>x</em>

Then <em>F</em> (0) = 5, <em>F</em> (0.1) ≈ 5.7, <em>F</em> (0.2) ≈ 6.4, <em>F</em> (0.5) ≈ 8.5, and <em>F</em> (2.0) ≈ 19. Notice how the error gets larger the further away <em>b </em>gets from 0.

A <em>better</em> numerical method would be Euler's method. Given <em>F'(x)</em>, we iteratively use the linear approximation at successive points to get closer approximations to the actual values of <em>F(x)</em>.

Let <em>y(x)</em> = <em>F(x)</em>. Starting with <em>x</em>₀ = 0 and <em>y</em>₀ = <em>F(x</em>₀<em>)</em> = 5, we have

<em>x</em>₁ = <em>x</em>₀ + 0.1 = 0.1

<em>y</em>₁ = <em>y</em>₀ + <em>F'(x</em>₀<em>)</em> (<em>x</em>₁ - <em>x</em>₀) = 5 + 7 (0.1 - 0)   →   <em>F</em> (0.1) ≈ 5.7

<em>x</em>₂ = <em>x</em>₁ + 0.1 = 0.2

<em>y</em>₂ = <em>y</em>₁ + <em>F'(x</em>₁<em>)</em> (<em>x</em>₂ - <em>x</em>₁) = 5.7 + 7.03 (0.2 - 0.1)   →   <em>F</em> (0.2) ≈ 6.403

<em>x</em>₃ = <em>x</em>₂ + 0.3 = 0.5

<em>y</em>₃ = <em>y</em>₂ + <em>F'(x</em>₂<em>)</em> (<em>x</em>₃ - <em>x</em>₂) = 6.403 + 7.12 (0.5 - 0.2)   →   <em>F</em> (0.5) ≈ 8.539

<em>x</em>₄ = <em>x</em>₃ + 1.5 = 2.0

<em>y</em>₄ = <em>y</em>₃ + <em>F'(x</em>₃<em>)</em> (<em>x</em>₄ - <em>x</em>₃) = 8.539 + 7.75 (2.0 - 0.5)   →   <em>F</em> (2.0) ≈ 20.164

4 0
2 years ago
Find the midpoint of points A (5,-6) and B(2,0) graphically​
netineya [11]
I think its (3.5, -3)
8 0
2 years ago
(1,-2),(-2,-5) find the slope and show me how u got it please​
stealth61 [152]
Answer: 1
y=1x-3
m=-3/-3=-1/-1=1
3 0
2 years ago
Read 2 more answers
Janie has 4 pieces of fruit. 2 of the pieces of fruit are bananas. Write 2 fractions that describe the fraction of fruit that ar
Aleonysh [2.5K]

Answer:

1/2

Step-by-step explanation:

4 fruit and 2 of them are banana so 2/4

banana go on top and total number of fruit go on bottom

to simplify divide numerator and denominator by 2

2/2          1

------ = ---------

4/2          2        

8 0
3 years ago
Read 2 more answers
Point G is between F and H. FH= 102, FG=5x+9, and HG=9x-5
dexar [7]

a)

FH = 102, FG = 5x + 9. HG = 9x - 5     <em>Given</em>

  FG    +   HG   =    FH                        <em>Segment addition postulate</em>

5x + 9 + 9x - 5 =  102                         <em>Substitution</em>

            14x + 4 = 102                        <em>Simplify (added like terms)</em>

                  14x = 98                          <em>Subtraction property of equality</em>

                     x = 7                             <em>Division property of equality</em>

b)   FG = 5x + 9  = 5(7) + 9   = 35 + 9   = 44

c)   HG = 9x - 5   = 9(7) - 5   = 63 - 5   = 58

Answers: a = 7, b = 44, c = 58

7 0
3 years ago
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