+1310 We connect dots with toothpicks in a grid as shown below. For example, the grid below has 7 horizontal toothpicks in each row and 5 vertical toothpicks in each column. \([asy] for (int i =0; i<=5; i+=1) { draw((0,i)--(7,i)); } for (int i=0; i<=7; i+=1) { draw((i,0)--(i,5)); for(int j=0; j<=5;j+=1) { dot((i,j)); } } [/asy] \)(a) Suppose we instead have a grid of dots that requires 10 horizontal toothpicks in each row and 20 vertical ones in each column. Then, how many total toothpicks will we need? Also, how many total dots are there? (b) Can you generalize your answer? Suppose we have a grid that requires $h$ horizontal toothpicks in each row and $v$ vertical toothpicks in each column. Then, how many total toothpicks will we need? Also, how many total dots are there? Be sure to include complete explanations with your answer, using complete sentences. Imagine you were going to show your solution to another person when you write it.
+128578 I don't see a picture, but I believe I know what you want.
7- horizontal in each row 5 - vertical in each column
The number of horizontal toothpicks =
(number of vertical toothpicks in each column + 1) * (number of horizontal toothpicks) =
( 6) * (7 ) = 42
The number of vertical toothpicks =
(number of horizontal toothpicks in each row + 1) * (number of vertical toothpicks) =
(8) * (5) = 40
So.....we need 42 + 40 = 82 toothpicks
Number of dots (nodes) =
(number of vertical toothpicks in each column + 1) * (number of horizontal toothpicks in each row + 1) = (6) * (8) = 48 dots (nodes)
With 10 horizontal and 20 wertical
# of horizontal toothpicks = (21)* 10 = 210
# of vertical toothpicks = (11) * 20 = 220
# of total toothpicks = 210 + 220 = 430
# of dots (nodes) = (21) * (11) = 231
We can extend this idea to any horizontal/vertical toothpick array.
Let h be the number of horizontal toothpicks and v the number of vertical
# of horizontal toothpicks = (v + 1) * h
# of vertical toothpicks = (h + 1) * v
# of total toothpicks = (v + 1) * h + (h + 1) * v
# of dots (nodal points) = (v + 1) * (h + 1)
