7. Law of cosines
a² = b² + c² - 2bc cos A
where a is the side opposite to ∠A and b and c are adjacent sides to ∠A.
Here, a=BC= 3.8
b=CA =4.8
c = AB = 3.2
Substituting the values in the formula,
3.8² = 4.8² + 3.2² - 2(4.8)(3.2) cos A
30.72 cos A =23.04 + 10.24 -14.44=18.84
cos A = 18.84/30.72 = 0.6133
∴ cos A =0.6133
8. It is a right angled triangle.
Pythagoras theorem
hypotenuse² = Perpendicular² + base²
Here, hypotenuse = 6.7, base = 5.4
Perpendicular² = 6.7² -5.4²=44.89 - 29.16 =15.73
Perpendicular = √15.73 = 3.966 = 3.97
∴ The length of the missing side is 3.97.
9. Given: Side of new cube = 5 * Side of original cube
Let the side of the original cube measure 1 unit.
So, volume of original cube = side³ = 1
The side of new cube = 5 * 1 =5 units
So, volume of new cube = side³ = 5³ =125
The volume of the bigger cube is 125 times larger than the original cube.
10. Given : Height of the cone = 7 in
Radius of the base of the cone = 2 in
Volume of cone = (1/3) πr²h
=(1/3)π(2²)(7)
=29.32 cubic inches
∴ Volume of the cone is 29.32 cubic inches.
11. The given equation is:
x² -18x +10 =0
(x² -18x + 9²) +10 -9² =0
(x-9)²-71 =0
(x-9)² =71
x-9 =√71
x = 9 + √71
∴ x = 9+√71
