pancha3

2020-10-18

Use function notation to describe the way the second variable (DV) depends upon the first variable (IV). Determine the domain and range for each, determine if there is a positive, negative, or no relationship, and explain your answers.
A)IV: an acute angle V in a right triangle: DV: the area B of the triangle if the hypotenuse is a fixed length G.
B)IV: one leg P of a right triangle: DV: the hypotenuse G of the right triangle if the other leg is 2
C)IV: the hypotenuse G of a right triangle: DV: the other leg P of the right triangle is one leg is 5.

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A) In this part, we are given an independent variable – an acute angle V in a right triangle, dependent variable - area B of the triangle and a constant – hypotenuse G has a fixed length.
We can first express the two legs of the right triangle in terms of hypotenuse G and acute angle V as shown below:
Height $=G\mathrm{sin}\left(V\right)$
Base $=G\mathrm{cos}\left(V\right)$
Use the formula for area of triangle as shown below:
Area $=\frac{1}{2}$ *Base*Height
Area $=\frac{1}{2}\cdot \left(G\mathrm{cos}\left(V\right)\right)\cdot \left(G\mathrm{sin}\left(V\right)\right)$
Area $=\frac{1}{2}{G}^{2}\mathrm{sin}\left(V\right)\mathrm{cos}\left(V\right)$
Area $=\frac{1}{4}{G}^{2}\mathrm{sin}\left(2V\right)$
Domain of this function is $\left(0,\frac{\pi }{2}\right)$ and range is $\left(0,\frac{{G}^{2}}{4}\right)$.
Since area function is oscillating function, therefore, the relation between two variables is neither positive and nor negative.
B) In this part, we are given an independent variable – a leg P of a right triangle, dependent variable - the hypotenuse G of the right triangle, and constant – second leg of length 2.
We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:
${P}^{2}+{2}^{2}$
${G}^{2}={P}^{2}+4$
$G=\sqrt{{P}^{2}+4}$
In reference to the given question, since P represents a leg of a right triangle, it can take any real number greater than 0. Therefore, domain of this function is $\left(0,\mathrm{\infty }\right)$.
Range of this function is all real numbers greater than 2, that is, $\left(2,\mathrm{\infty }\right)$.
Since value of hypotenuse increases as the length of leg increases, therefore, there is a positive relationship between the two variables.
C) In this part, we are given an independent variable – the hypotenuse G of a right triangle, dependent variable - the leg P of the right triangle, and constant – second leg of length 5.
We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:
${P}^{2}+{5}^{2}={G}^{2}$
${P}^{2}={G}^{2}-25$
$P=\sqrt{{G}^{2}-25}$
G can take any real number greater than 5 in order for this function to exist. Therefore, domain of this function is $\left(5,\mathrm{\infty }\right)$.
Range of this function is all positive real numbers, that is, $\left(0,\mathrm{\infty }\right)$.
Since value of leg increases as the length of hypotenuse increases, therefore, there is a positive relationship between the two variables.

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