The increase in length of an aluminum rod is twice the increase in length of an Invar rod with only a third of the temperature increase. Find the ratio of the lengths of the two rods.

the ratio of lengths of the two rods, Aluminum to Invar is 11.27

Step-by-step explanation:

coefficient of linear expansion of aluminum,

Coefficient of linear expansion of Invar,

Linear thermal expansion is given as;

The increase in length of Invar is given as;

The increase in length of the Aluminum;

Therefore, the ratio of lengths of the two rods, Aluminum to Invar is 11.27

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)