Mathematics: 30 members of a local gym were randomly selected to answer a survey. The selected gym members were between the ages of 50 and 65. Does that mean that all of the gym members are between the ages of 50 and 65?

Equation of hyperbola is given byWhere the centre is (h, k) and the foci is (h-c, k) and (h+c, k)c is obtained from We ll calculate the distance for each option in turnOption AThe centre is (2, 1)Work out the value of c to find the coordinates of two focusWe have and The coordinate of foci is and Notice that the y-coordinate of the focus is the same with the y-coordinate of the centre, so we ll only need to work out the horizontal distance between one foci to the centre (the centre of a hyperbola...

30 members of a local gym were randomly selected

Ответ:

shane9564

22.11.2021

Mathematics

Equation of hyperbola is given by
$\frac{(x-h)^2}{a^2}- \frac{(y-k)^2}{b^2}=1$

Where the centre is (h, k) and the foci is (h-c, k) and (h+c, k)

c is obtained from c^2=a^2+b^2

We'll calculate the distance for each option in turn

Option A

$\frac{(x-2)^2}{8^2}- \frac{(y-1)^2}{7^2}=1$

The centre is (2, 1)

Work out the value of 'c' to find the coordinates of two focus
We have a^2=8^2

and

$c= \sqrt{113}$

The coordinate of foci is $(2- \sqrt{113}, 1)$ and $(2+ \sqrt{113}, 1)$

Notice that the y-coordinate of the focus is the same with the y-coordinate of the centre, so we'll only need to work out the horizontal distance between one foci to the centre (the centre of a hyperbola is the same distance to both focus)

Distance from foci to centre = $(2+ \sqrt{113})-2=12.6-2=10.6$
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Option B

Centre (-2, 3)
c² = a² + b² = 19² + 11² = 482
c = √482
c = 21.95

The coordinates of focus = (-2+21.95, 3) and (-2-21.95, 3)
= (19.95, 3) and (-23.95, 3)

Distance from one foci to the centre is = 19.95 - (-2) = 21.95 units
If the other foci is used for the calculation, the answer will be the same
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Option C

Centre (1, 2)
c² = 6² + 9² = 36 + 81 = 117
c = √117
c = 10.8

Coordinates of focus = ((1-10.8), 2) and ((1+10.8), 2)
= (-9.8, 2) and (11.8, 2)

Distance between foci and centre = 11.8 - 1 = 10.8 units
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Option D

Centre (5, -3)

c² = 5² + 19² = 386
c = √386
c = 19.6

Coordinates of focus are ((5-19.6), -3) and ((5+19.6), -3) = (-14.6, -3) and (24.6, -3)

Distance foci - centre = 24.6 - 5 = 19.6
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The equation that gives the furthest distance from centre to foci is given by option B

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30 members of a local gym were randomly selected to answer a survey. The selected gym members were between the ages of 50 and 65. Does that mean that all of the gym members are between the ages of 50 and 65?

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