A 9 ft by 8 ft bookcase is to be built. It has horizontal shelves every foot. A support is to be notched in the shelves diagonally from one corner to the opposite corner. At what point should each of the shelves be notched? That is, find lengths A, B, C, D and E in the illustration. How long is the diagonal crosspiece? Round your answer to the nearest hundredth.
A) A = 1.3 ft, B = 2.7 ft, C = 4 ft, D = 5.3 ft,E = 6.7 ft, the crosspiece is 12 ft.
B) A = 1.5 ft, B = 3 ft, C = 4.5 ft, D = 6 ft,E = 7.5 ft, the crosspiece is 12.04 ft.
C) A = 1.5 ft, B = 3 ft, C = 5.5 ft, D = 5 ft,E = 7.5 ft, the crosspiece is 12.04 ft.
D) A = 1.33 ft, B = 2.67 ft, C = 4 ft, D = 5.33 ft,E = 6.67 ft, the crosspiece is 12.04 ft.
E) A = 1.5 ft, B = 3 ft, C = 4.5 ft, D = 6 ft,E = 6.5 ft, the crosspiece is 12 ft.

Question

A 9 ft by 8 ft bookcase is to be built. It has horizontal shelves every foot. A support is to be notched in the shelves diagonally from one corner to the opposite corner. At what point should each of the shelves be notched? That is, find lengths A, B, C, D and E in the illustration. How long is the diagonal crosspiece? Round your answer to the nearest hundredth. 
A) A = 1.3 ft, B = 2.7 ft, C = 4 ft, D = 5.3 ft,E = 6.7 ft, the crosspiece is 12 ft. 
B) A = 1.5 ft, B = 3 ft, C = 4.5 ft, D = 6 ft,E = 7.5 ft, the crosspiece is 12.04 ft. 
C) A = 1.5 ft, B = 3 ft, C = 5.5 ft, D = 5 ft,E = 7.5 ft, the crosspiece is 12.04 ft. 
D) A = 1.33 ft, B = 2.67 ft, C = 4 ft, D = 5.33 ft,E = 6.67 ft, the crosspiece is 12.04 ft. 
E) A = 1.5 ft, B = 3 ft, C = 4.5 ft, D = 6 ft,E = 6.5 ft, the crosspiece is 12 ft.

Quizplus · Accepted Answer

To find the length of the diagonal crosspiece, we can use the Pythagorean theorem. Let's find the length of the longest diagonal first:

$h = \sqrt{9^2 + 8^2} \approx 12.04$

Each notch cuts off a right triangle with legs A and B, and hypotenuse h. Since the triangles are similar, we can set up the ratios:

$A/h = 1/12$
$B/h = 2/12$
$C/h = 3/12$
$D/h = 4/12$
$E/h = 5/12$

Solving for each notch length:

$A = (1/12)h \approx 1.5$
$B = (2/12)h \approx 3$
$C = (3/12)h \approx 4.5$
$D = (4/12)h \approx 6$
$E = (5/12)h \approx 7.5$

Therefore, the best choice is B, with A = 1.5 ft, B = 3 ft, C = 4.5 ft, D = 6 ft,E = 7.5 ft, and the diagonal crosspiece length of approximately 12.04 ft.

A 9 Ft by 8 Ft Bookcase Is to Be