Let the width of rectangle MATH equal w. The length of the rectangle is 3cm longer than the width. In rectangle GEOM the width is 1 more than w and the length is 5 less than twice w. If their perimeters are equal, find w.

7 cm

Step-by-step explanation:

The perimeter P of a rectangle is given as

p = 2 (l + b) where l and b are the length and width respectively

hence for a rectangle where  the length of the rectangle is 3cm longer than the width w

the perimeter

= 2 (w + 3 + w)

= 2(2w + 3)

In another rectangle where the width is 1 more than w and the length is 5 less than twice w, the perimeter

= 2(2w - 5 + w +1)

= 2 (3w - 4)

since the perimeters are equal, equate both equations

2(2w + 3) = 2 (3w - 4)

2w + 3 = 3w -4

collect like terms

3w - 2w = 4 + 3

w = 7cm

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺