Triangular numbers can be represented with equilateral triangles formed by dots. The first five triangular numbers are 1, 3, 6, 10, and 15. Is there a direct variation between a triangular number and its position in the sequence? Explain your reasoning.

Answers

Answer 1

Answer: Hello,

$T_(1) =1\n T_(2) =3=1+2\n T_(3) =6=1+2+3\n T_(4) =10=1+2+3+4\n T_(5) =15=1+2+3+4+5\n...\n T_(n) =1+2+3+....+n= (n*(n+1))/(2) \n$

Answer 2

Answer:

Answer: No, the triangular numbers are not a direct variation. There is not a constant of variation between a number and its position in the sequence. The ratios of the numbers to their positions are not equal. Also, the points (1, 1), (2, 3), (3, 6), and so on, do not lie on a line.

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Rosa read 10 pages of a book on monday. She read 1/3 of the remainder on tuesday. If she still had 24 pages to read, how many pages were there in the book

Answers

The total pages in the book are 46.

What is equation?

An equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Given that, Rosa read 10 pages of a book on Monday. She read 1/3 of the remainder on Tuesday, and she still had 24 pages to read,

Let the total number of pages be x,

Establishing the equation, we get,

10+(x-10) / 3+24 = x

30 + x - 10 + 72 = 3x

92 = 2x

x = 46

Hence, The total pages in the book are 46.

For more references on equations, click;

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Answer:

There are 46 pages in the book.

Step-by-step explanation:

if she already read 10 pages of the book and the remainder is split into 3rds and she already read 1/3 of the book then the 24 pages get divided by the 2/3 left which equals 12 so each third is 12 pages so 12 x 3 = 36 + 10= 46

There are 46 pages in the book.

a paddle boat can move at a speed of 6km/h in still water. the boat is paddeled 16km downstream in a river in the same time it takes to go 8km upstream what is the speed of the river?

Answers

$v-the\ speed\ of\ the\ river\n\n16:(6+v)=8:(6-v)\n\n(16)/(6+v)=(8)/(6-v)\n-------------\nmultiply\ on\ the\ cross\n-------------\n8(6+v)=16(6-v)\n\n48+8v=96-16v\n\n8v+16v=96-48\n\n24v=48\ \ \ /:24\n\nv=2\ (km/h)\leftarrow Answer$

Find the zeros in simplest radical form:
y=1/2x^2-4

Answers

$y= (1)/(2)x^2-4\n \n y =0 \n \n(1)/(2)x^2-4 =0 \ \ / \cdot 2\n \nx^2-8=0 \n \n(x-√(8))(x+√(8))=0 \n \n x-√(8)= \ \ or \ \ x+√(8) = 0 \n \nx=√(8) \ \ or \ \ x=-√(8) \n \nx=√(4\cdot 2) \ \ or \ \ x= -√(4\cdot 2)\n \n x=2√(2) \ \ or \ \ x=-2√(2)$

$y= (1)/(2) x^2-4\n\ny=0\ \ \ \Leftrightarrow\ \ \ (1)/(2) x^2-4=0\ /\cdot2\ \ \ \Leftrightarrow\ \ \ x^2-8=0\n\nx^2-(2 √(2) )^2=0\ \ \ \Leftrightarrow\ \ \ (x-2 √(2) )(x+2 √(2) )=0\n\nx-2 √(2) =0\ \ \ \ \ \ \ \ or\ \ \ \ \ \ \ \ \ x+2 √(2)=0\n\nx=2 √(2)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=-2 \sqrt{2$

if the equation is h= -2x^2 + 12x -10

how do I find the max height?

Answers

One other way to solve this question is finding the derivative

$h=-2x^2+12x-10$

$h'=-4x+12$

now we have to find when this function will be zero

$-4x+12=0$

$\boxed{\boxed{x=3}}$

now we just replace this value at our initial function

$h=-2x^2+12x-10$

$h_(max)=-2*(3)^2+12*3-10$

$h_(max)=-18+36-10$

$\boxed{\boxed{h_(max)=8}}$

The maximum height is the ordinate value of the vertex of the parabola, ie: yV

Calculating yV:

$y_V=(-\Delta)/(4a)\n \n y_V=-[(12^2-4*(-2)*(-10)])/(4*(-2))=(-(144-80))/(-8)=(-64)/(-8)=8$

Please Help!! 9 points ! brainliest answer

Answers

Answer:

Answer C I think

Step-by-step explanation:

Find the value of M3 - M1

Answers

Answer:

Step-by-step explanation:

m<1+m<2+m<3 = 180 this forms a straight line

m<2 = 90 degrees

m<1 + 90 + m<3 = 180

Subtract 90 from each side

m<1 + 90-90 + m<3 = 180-90

m<1 + m<3 = 90

70 +m<1+m<2 = 180 this forms a straight line

70 + m<1 + 90 = 180

Combine like terms

160 + m<1 = 180

Subtract 160 from each side

160-160 + m<1 = 180-160

m<1 = 20

m<1 + m<3 = 90

20 + m<3 = 90

Subtract 20 from each side

20-20 +m<3=90=20

m<3 = 70

We want to find m<3 - m<1

m<3 - m<1

70 -20

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