(a) The curved surface areas of two cones are equal. The base radius of one is 5 cm and its slant height is 12cm. calculate the height of the second cone if its base radius is 6 cm.
(b) Given the matrices A = \(\begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix}\) and B = \(\begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix}\), find:
a) We can use the formula for the curved surface area of a cone, which is given by πrs, where r is the radius of the base and s is the slant height. Let's assume that the height of the second cone is h. Then we have:
π(5)(12) = π(6)(√(6^2 - h^2))
Simplifying the equation, we get:
60 = 6√(36 - h^2)
10 = √(36 - h^2)
Squaring both sides, we get:
100 = 36 - h^2
h^2 = 64
h = 8
Therefore, the height of the second cone is 8 cm.
b) (i) To find BA, we need to multiply the matrices in the order B x A. The product is given by:
BA = ∡(3 -2) (4 1) x (2 5) (-1 -3) = ∡(8 21) (7 17)
(ii) The determinant of a product of two matrices is equal to the product of their determinants, i.e., |AB| = |A||B|. Therefore, to find the determinant of BA, we can multiply the determinants of B and A. The determinant of a 2x2 matrix is given by the formula ad - bc, where a, b, c, and d are the elements of the matrix. Using this formula, we get:
|BA| = (8)(17) - (21)(7) = -49
Therefore, the determinant of BA is -49.