ANSWER
The length is 10 inches and the width is 24 inches
EXPLANATION
The diagonal of the rectangular painting is d=26 inches.
Let l and w be the length and width of the painting respectively.
From Pythagoras Theorem,
[tex] {l}^{2} + {w}^{2} = {26}^{2} [/tex]
[tex]{l}^{2} + {w}^{2} = 676..(1)[/tex]
Its area is 240 square inches.
This implies that,
[tex]l \times w = 240[/tex]
[tex]l = \frac{240}{w} ...(2)[/tex]
Put equation (2) into (1).
[tex]{( \frac{240}{w} )}^{2} + {w}^{2} = 676[/tex]
This implies that,
[tex] {w}^{4} - 676 {w}^{2} + 57600 = 0[/tex]
[tex]( {w}^{2} - 100)( {w}^{2} - 576) = 0[/tex]
[tex]{w}^{2} - 100 = 0 \: or \: ({w}^{2} - 576= 0[/tex]
[tex]{w}^{2} = 100 \: or \: {w}^{2} = 576[/tex]
Take positive square root to get,
[tex]{w} = 10\: or \: {w} = 24[/tex]
When w=24,
[tex]l = \frac{240}{24} = 10[/tex]
when w=10
[tex]l = \frac{240}{10} = 24[/tex]
Hence the length is 10 inches and the width is 24 inches.
