Question

areas of four triangles are 40 cm^2, 90 cm^2, 202.5 cm^2 and 455.6 cm^2 are continue in sequence. the enlargement factor is 1.5, which triangle have first area of 15000 cm^2

Answer 1

9th Triangle

Explanation:

Given the area of four triangles:
`40 cm^2, 90 cm^2, 202.5 cm^2 \:and\: 455.6 cm^2`

If the Scale factor is 1.5.

This is a geometric sequence and the common ratio is 1.5 X 1.5.

Therefore, the area of any n triangle can be determined using the function:

`A(n)=40\cdot1.5^(2(n-1))`

We want to determine which triangle has an area greater than
`15000cm^2`.

When A(n)=15000

`40\cdot1.5^(2(n-1))>15000\\1.5^(2(n-1))>(15000)/(40) \\1.5^(2(n-1))>375\\\text{Change to Logarithm form}\\2(n-1)>Log_(1.5)375\\\text{Applying logarithm change of base to base 10 law}\\2(n-1)>(Log 375)/(Log 1.5) \\2n-2>14.62\\2n>14.62+2=16.62\\n>8.31`

Therefore, the 9th triangle has an area greater than
`15000cm^2`.