Question

Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal.
1. Use the table and the method of this example to find an equation that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute).
weight, X 25 35 50 75 500 1000
Galloping speed, Y 191.5 184.7 175.8 162.2 124.9 113.2

Answer 1

Answer:

Galloping speed Y=179.984-0.076 weight X

Step-by-step explanation:

The required equation is

y=a+bx

where y is galloping speed and x is weight of animal.

We have to estimate intercept a and slope b.

x                  y         xy       x²

25    191.5 4787.5 625

35     184.7 6464.5 1225

50    175.8 8790 2500

75     162.2 12165 5625

500  124.9 62450 250000

1000 113.2 113200 1000000

sumx=1685

sumy=952.3

sumxy=207857

sumx²=1259975

$slope=b=\frac{n(sumxy)-(sumx)(sumy)}{nsumx^{2}-(sumx)^{2} }$

$slope=b=\frac{6(207857)-(1685)(952.3)}{6*(1259975)-(1685)^{2} }$

$slope=b=\frac{-357483.5}{4720625}$

slope=b=-0.07573.

intercept=a=ybar-b*xbar

$ybar=\frac{sumy}{n}$

$ybar=\frac{952.3}{6}$

ybar=158.7167

$xbar=\frac{sumx}{n}$

$xbar=\frac{1685}{6}$

xbar=280.8333

intercept=a=ybar-b*xbar

intercept=a=158.7167-(-0.07573)*280.8333

intercept=a=158.7167+21.2675

intercept=a=179.9842

rounding intercept and slope to 3 decimal places

intercept=179.984.

slope=-0.076

The required equation is

Galloping speed Y=179.984-0.076 weight X

Answer 2

The required equation is  galloping speed Y = 179.984 - 0.076 weight X.

Given that,

Four-legged animals run with two different types of motion: trotting and galloping.

The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal.

We have to determine,

An equation that relates an animal's weight x (in pounds) and its lowest galloping speed y (in strides per minute).

According to the question,

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.

The required linear equation is,

y = a + bx

Where y is galloping speed and x is weight of animal.

To estimate intercept a and slope b from the table showing below.

x        y         x.y           x²

25     191.5   4787.5   625

35     184.7   6464.5  1225

50     175.8   8790     2500

75      162.2  12165     5625

500   124.9   62450  250000

1000  113.2   113200  1000000

Then,

sum of x = 1685 , sum of y = 952.3 , sum of xy = 207857 , sum of x² = 1259975.

Therefore, Slope of b is given by the formula;

$b = \dfrac{(sum\ of \ xy )- (sumx).(sumy)}{n(sumx^{2})- (sumx)^{2} }\\\\b = \dfrac{6(207857)-1685(952.3)}{6(1259975)-(1685)^{2} }\\\\b = \dfrac{-357483.5}{4720625}\\\\b = -0.7573$

Then,

$Y_b_a_r = \dfrac{sumx}{n} = \dfrac{952.3}{6} = 158.71\\\\X_b_a_r = \dfrac{sumy}{n} = \dfrac{1685}{6} = 280.83$

Therefore, The intercept is given by,

$intercept = a = Y_b_a_r-b\times X_b_a_r\\\\intercept = a = 158.7167 - (-0.07573)\times 280.8333\\\\intercept = a = 158.7167 + 21.2675\\\\intercept = a = 179.984$

On rounding intercept and slope to 3 decimal places,

intercept = 179.984.

slope = -0.076

Hence, The required equation is  galloping speed Y = 179.984 - 0.076 weight X.

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