Question

1. the table below shows the lengths and corresponding ideal weights of sand sharks.

length weight
60 107
62 116
64 126
66 135
68 139
70 152
predict the weight of a sand shark whose length is 75 inches.
if the weight of the shark is 200, what is the predicted length?
equation ________
weight expected ________
length ________

Explanation:

Step1: Assume linear - relationship

Let the relationship between length $L$ and weight $W$ be $W = aL + b$. We can use the method of least - squares or two - point form to find the equation of the line. First, take two points from the table, say $(L_1,W_1)=(60,107)$ and $(L_2,W_2)=(62,116)$.
The slope $a=\frac{W_2 - W_1}{L_2 - L_1}=\frac{116 - 107}{62 - 60}=\frac{9}{2}=4.5$.
Using the point - slope form $W - W_1=a(L - L_1)$ with $(L_1,W_1)=(60,107)$ and $a = 4.5$, we get $W-107 = 4.5(L - 60)$.
Expanding, $W-107=4.5L-270$, so $W = 4.5L - 163$.

Step2: Predict weight for $L = 75$

Substitute $L = 75$ into the equation $W = 4.5L-163$.
$W=4.5\times75 - 163$.
$W = 337.5-163$.
$W = 174.5$.

Step3: Predict length for $W = 200$

Set $W = 200$ in the equation $W = 4.5L-163$.
$200=4.5L-163$.
Add 163 to both sides: $200 + 163=4.5L$, so $363 = 4.5L$.
Then $L=\frac{363}{4.5}=\frac{3630}{45}=\frac{242}{3}\approx80.67$.

Answer:

The weight of a sand - shark with length 75 inches is 174.5. The length of a sand - shark with weight 200 is approximately 80.67 inches.