Question

A fireplace arch is to be constructed in the form of a semiellipse. The opening is to have a height of 3 feet at the center and a width of 8 feet along the base. The contractor cuts a string of a certain length and nails each end of the string along the base in order to sketch the outline of the semiellipse.a.) What is the total length of the string?b.) How far from the center should the string be nailed into the base?

Answer 1

Answer:

Step-by-step explanation:

Given

Fireplace is in the form of semi-ellipse  with height of 3 ft from center

Length of base=8 ft

if the arch is in the form of ellipse then it can be considered as horizontal ellipse with 2 a and 2 b be length of major and minor axis

2 a =8

a=4 ft

If contractor cover the whole base

then total length of string=2a =8 ft

(b)String should be nailed at focus

focus of ellipse is a e

where e=eccentricity

$e=\sqrt{1-(\frac{b}{a})^2}$

$e=\sqrt{1-(\frac{3}{4})^2}=\frac{\sqrt{7}}{4}$

Focus $=ae =\sqrt{7}$

Thus string should be nailed at 2.64 m from center

Answer 2

The distance between two points is the number of units between them.

• The total string length is 8ft
• The string should be positioned at a distance of $\mathbf{\sqrt 7}$ from the center

The given parameters are:

$\mathbf{h= 3}$

$\mathbf{w= 8}$

(a) The total string length

The total string length is the length along the base.

Hence, the total string length is 8ft

(b) How far from the center should the string be nailed

The position of the string should be at the focus (f).

This is calculated as:

$\mathbf{f =ae}$

Where:

$\mathbf{a = \frac 12 \times 8 = 4}$ ---- 1/2 the length of the string

$\mathbf{e = \sqrt{1 - (\frac ba)^2}}$ ---- eccentricity

Where

$\mathbf{b = h = 3}$

So, we have:

$\mathbf{e = \sqrt{1 - (\frac{3}{4})^2}}$

$\mathbf{e = \sqrt{1 - \frac{9}{16}}}$

$\mathbf{e = \sqrt{ \frac{7}{16}}}$

So, we have:

$\mathbf{e = \frac{\sqrt 7}{4}}$

The focus is then calculated as:

$\mathbf{f = 4 \times \frac{\sqrt 7}{4} }$

$\mathbf{f = \sqrt 7}$

Hence, the string should be positioned at a distance of $\mathbf{\sqrt 7}$ from the center.

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