Mathematics: Using pythagorean inequalities, determine which of the following sets of values can represent the lengths of the sides of an acute triangle. a. 1,2,3 b. 6,8,10 c. 6,7,8 d. 4,6,10

Using pythagorean inequalities, determine which

Ответ:

samueltaye

13.01.2022

c. 6, 7, 8

Step-by-step explanation:

$a,\ b,\ c-\text{sides of a triangle}\\\\\text{If}\ a\leq b< c\ \text{and}\ a^2+b^2=c^2,\ \text{then it's the lengths of the sides of a right triangle}\\\\\text{If}\ a\leq b\leq c\ \text{and}\ a^2+b^2c^2,\ \text{then it's the lengths of the sides of a acute triangle}\\\\\text{If}\ a\leq b< c\ \text{and}\ a^2+b^2$

$\text{Triangle condition:}\\\text{That each side has to be shorter than the sum of the other two sides}\\\text{and longer than their difference.}\\\\\text{If}\ a\leq b\leq c,\ \text{then}\ a+bc.$

$--------------------------\\a.\\a=1,\ b=2,\ c=3\\\\a+b=1+2=3=c\\\\\text{These are not the lengths of the sides of the triangle.}\\----------------------------\\b.\\a=6,\ b=8,\ c=10\\\\a+b=6+8=1410=c\qquad OK\\\\a^2+b^2=6^2+8^2=36+64=100\\\\c^2=10^2=100\\\\a^2+b^2=c^2\\\\\text{It's the right triangle.}\\----------------------------$

$c.\\a=6,\ b=7,\ c=8\\\\a+b=6+7=138\qquad OK\\\\a^2+b^2=6^2+7^2=36+49=85\\\\c^2=8^2=64\\\\a^2+b^2c^2\\\\\text{It's the acute triangle.}\\----------------------------\\d.\\a=4,\ b=6,\ c=10\\\\a+b=4+6=10=c\\\\\text{These are not the lengths of the sides of the triangle.}$

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Ответ:

mjam85877

30.11.2021

Mathematics

$\large \boxed{\text{Part A. 1280; Part B. 1320; Part C; 1333}}$

Step-by-step explanation:

Part A. Using five rectangles.

I will use the left end-point approximation.

1. Divide the interval [0,20] into five equal subintervals

Δx = (20 - 0)/5 = 4

The intervals are:

[0,4], [4,8], [8,12], [12,16], [16,20]

2. Evaluate the function at the left end-points and calculate the areas

$\begin{array}{rrr}\mathbf{x} & \mathbf{f (x)}&\textbf{Area} \\0 &0&0 \\4 & 64&256 \\8 & 96&384 \\12 & 96&384 \\16 & 64&256 \\&\textbf{TOTAL =}& \mathbf{1280}\end{array}\\\text{The area of five rectangles is $\large \boxed{\mathbf{1280}}$}$

Fig. 1 shows the graph with five rectangles.

Part B. Using 10 rectangles

1. Divide the interval [0,20] into 10 equal subintervals

Δx = (20 - 0)/10 = 2

The intervals are:

[0,2], [2,4], [4,6], [6,8], [8,10], [10,12], [12,14], [14,16], [16,18], [18,20]

2. Evaluate the function at the left end-points and calculate the areas

$\begin{array}{rrr}\mathbf{x} & \mathbf{f (x)}&\textbf{Area} \\0 &0&0 \\2 & 36&72 \\4 & 64&128 \\6 & 84&168 \\8 & 96&192 \\10 &100&200 \\12 & 96&192 \\14 &84&168 \\16 & 64&128 \\18 &36&72 \\&\textbf{TOTAL =}& \mathbf{1320}\end{array}\\\text{The area of 10 rectangles is $\large \boxed{\mathbf{1320}}$}$

Fig. 2 shows the graph with 10 rectangles.

Part C. The limit as n ⟶ ∞

$\displaystyle \lim_{n\longrightarrow \,\infty} \sum_{i = 1}^{n} f(x_{i-1})\Delta x= \int_{a}^{b}f(x)dx$

Integrate the function.

$\displaystyle \int_{0}^{20}(20x-x^{2})dx= \left[10x^{2}- \dfrac{1}{3}x^{3}\right]_{0}^{20} =4000 - \dfrac{8000}{3} = \dfrac{12000 - 8000 }{3}\\\\ = \dfrac{4000}{3} \approx\mathbf{1333}$

I NEED THIS ASAP In this task, you will practice finding the area under a nonlinear function by usin

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Using pythagorean inequalities, determine which of the following sets of values can represent the lengths of the sides of an acute triangle.

a. 1,2,3
b. 6,8,10
c. 6,7,8
d. 4,6,10

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