Tell whether each relation is a function. {(2, 5), (3, 5), (4, −4 ), (5, −4 ), (6, 8)}
2 answers:
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If you look at the pyramid, the pyramid is really just a hexagon with 6 triangles stacked up to each other at the edges, with the hexagon at the bottom.
now, the perpendicular distance from the center of the hexagon to a side, namely the apothem, is 6√3, and each side is 12 units long, since there are 6 of them that'd be 72 for all, namely the perimeter.
each of the triangular faces have a base of 12 and an altitude or height of 11, recall that area of a triangle is (1/2)bh, and area of a regular polygon is (1/2)(apothem)(perimeter).
so if we just get the area of the hexagon at the bottom, and the triangles, sum them up, that's the surface area of the pyramid.
![\bf \stackrel{\textit{area of the hexagon}}{\left[\cfrac{1}{2}(6\sqrt{3})(72) \right]}~~~~+~~~~\stackrel{\textit{area of the 6 triangles}}{6\left[\cfrac{1}{2}(12)(11) \right]}](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7B%5Ctextit%7Barea%20of%20the%20hexagon%7D%7D%7B%5Cleft%5B%5Ccfrac%7B1%7D%7B2%7D%286%5Csqrt%7B3%7D%29%2872%29%20%20%5Cright%5D%7D~~~~%2B~~~~%5Cstackrel%7B%5Ctextit%7Barea%20of%20the%206%20triangles%7D%7D%7B6%5Cleft%5B%5Ccfrac%7B1%7D%7B2%7D%2812%29%2811%29%20%20%5Cright%5D%7D)
now, the perpendicular distance from the center of the hexagon to a side, namely the apothem, is 6√3, and each side is 12 units long, since there are 6 of them that'd be 72 for all, namely the perimeter.
each of the triangular faces have a base of 12 and an altitude or height of 11, recall that area of a triangle is (1/2)bh, and area of a regular polygon is (1/2)(apothem)(perimeter).
so if we just get the area of the hexagon at the bottom, and the triangles, sum them up, that's the surface area of the pyramid.
In both cases there are more than one possible function sutisfying given data.
1. If
- x‑intercepts are (–5, 0), (2, 0), and (6, 0);
- the domain is –5 ≤ x ≤ 7;
- the range is –4 ≤ y ≤ 10,
then (see attached diagram for details) you can build infinetely many functions. From the diagram you can see two graphs: first - blue graph, second - red graph. Translating their maximum and minimum left and right you can obtain another function that satisfies the conditions above.
2. If
- x‑intercepts are (–4, 0) and (2, 0);
- the domain is all real numbers;
- the range is y ≥ –8,
then you can also build infinetely many functions. From the diagram you can see two graphs: first - blue graph, second - red graph. Translating their minimum left and right you can obtain another function that satisfies the conditions above.
Note, that these examples are not unique, you can draw a lot of different graphs of the functions.
Answer: yes, there are more than one possible function
use
to get
then use
and
also,
to get
then again use the first identity In both pairs, i.e.
to get
multiply and divide by 4 to get the RHS.
because,
squaring both sides,