Question

A rare whale fossil was uncovered under layers of sediment. When the excavator took measurements, she found the tail to be as long as its head plus a quarter of the length of the body. When the archaeologist came in, he measured the length of the body. He found it to be ¾ of the total length. They made arguments about how such a fish would swim in the water. To make proper predictions they measured the head size, and found it to be 4 inches long. What is the total length of this whale?

128

99

136

142

with proof

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Answer 1

Answer :-

Let us assume tail to be t, head to be h, length of body be x and the total length be L.

Tail = Head + ¼ of length of body

→ t = h + ¼ x

Length of body = ¾ of total length

→ x = ¾ L

Also,

Total length = Length of body + Head + tail

→ L = x + h + t

Now, the question has given us the length of head. Substituting the value :-

$\implies\sf t = 4 + \dfrac{1}{4}x \: \: \: \: \: \: -i$

$\implies\sf x = \dfrac{3}{4} L \: \: \: \: \: \: -ii$

$\implies\sf L = x + 4 + t \: \: \: \: \: \: -iii$

Substituting the value of t from equation i in equation iii :-

$\implies\sf L = x + 4 + t$

$\implies\sf L = x + 4 + 4 + \dfrac{1}{4}x$

$\implies\sf L = 8 + x + \dfrac{1}{4}x\: \: \: \: \: \: -iv$

Substituting the value of x from equation ii in equation iv :-

$\implies\sf L = 8 + \dfrac{3}{4} L + \dfrac{1}{4} \times \dfrac{3}{4} L$

$\implies\sf L = 8 + \dfrac{3}{4} L + \dfrac{3}{16} L$

$\implies\sf L = 8 + \dfrac{12}{16} L + \dfrac{3}{16} L$

$\implies\sf L = 8 + \dfrac{15}{16} L$

$\implies\sf L - \dfrac{15}{16} L = 8$

$\implies\sf \dfrac{1}{16} L = 8$

$\implies\sf L = 8 \times 16$

$\implies\sf L = 128$

Verification :-

From equation ii :-

$\implies\sf x = \dfrac{3}{4}L$

$\implies\sf x = \dfrac{3}{4} \times 128$

$\implies\sf x = 96$

Substituting the value in equation i :-

$\implies\sf t = h + \dfrac{1}{4} x$

$\implies\sf t = 4 + \dfrac{1}{4} \times 96$

$\implies\sf t = 4 + 24$

$\implies\sf t = 28$

Substituting the value of t , x , h and L in equation iii :-

$\implies\sf L = t + x + h$

$\implies\sf LHS \longrightarrow L = 128$

$\implies\sf RHS \longrightarrow t + x + h$

$\implies\sf RHS = 28 + 96 + 4$

$\implies\sf RHS = 128$

LHS = RHS

Hence verified.

Answer 2

Given :  tail to be as long as its head plus a quarter of the length of the body

length of the body be ¾ of the total length.

Head 4 inches long

To Find :   total length of this whale

128

99

136

142

Solution:

Head = 4 inch

Body  = B inch

Tail =  Head + (1/4) Body

= 4 + (1/4)B

= 4 + B/4   inches

Total Length =  Head + Body + Tail

= 4 + B  + 4 + B/4

= 8 + 5B/4

Body = (3/4) Total Length

=> B  = (3/4)(8 + 5B/4)

=> 4B  = 3( 32 + 5B)/4

=> 16B = 96 + 15B

=> B = 96

Total Length = 8 + 5B/4  = 8 + 5 * 96/4

= 8 + 5 * 24

= 8 + 120

= 128  inches

Head = 4 inches

Body  = 96  inches

Tail  = 28  inches

Total Length = 128 inches

total length of this whale 128 inches

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