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.

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.

Question
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.

Answers (1)

2020-10-19
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 \(\displaystyle={G}{\sin{{\left({V}\right)}}}\)
Base \(\displaystyle={G}{\cos{{\left({V}\right)}}}\)
Use the formula for area of triangle as shown below:
Area \(\displaystyle=\frac{{1}}{{2}}\) *Base*Height
Area \(\displaystyle=\frac{{1}}{{2}}\cdot{\left({G}{\cos{{\left({V}\right)}}}\right)}\cdot{\left({G}{\sin{{\left({V}\right)}}}\right)}\)
Area \(\displaystyle=\frac{{1}}{{2}}{G}^{{2}}{\sin{{\left({V}\right)}}}{\cos{{\left({V}\right)}}}\)
Area \(\displaystyle=\frac{{1}}{{4}}{G}^{{2}}{\sin{{\left({2}{V}\right)}}}\)
Domain of this function is \(\displaystyle{\left({0},\frac{\pi}{{2}}\right)}\) and range is \(\displaystyle{\left({0},\frac{{G}^{{2}}}{{4}}\right)}\).
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:
\(\displaystyle{P}^{{2}}+{2}^{{2}}\)
\(\displaystyle{G}^{{2}}={P}^{{2}}+{4}\)
\(\displaystyle{G}=\sqrt{{{P}^{{2}}+{4}}}\)
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 \(\displaystyle{\left({0},\infty\right)}\).
Range of this function is all real numbers greater than 2, that is, \(\displaystyle{\left({2},\infty\right)}\).
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:
\(\displaystyle{P}^{{2}}+{5}^{{2}}={G}^{{2}}\)
\(\displaystyle{P}^{{2}}={G}^{{2}}-{25}\)
\(\displaystyle{P}=\sqrt{{{G}^{{2}}-{25}}}\)
G can take any real number greater than 5 in order for this function to exist. Therefore, domain of this function is \(\displaystyle{\left({5},\infty\right)}\).
Range of this function is all positive real numbers, that is, \(\displaystyle{\left({0},\infty\right)}\).
Since value of leg increases as the length of hypotenuse increases, therefore, there is a positive relationship between the two variables.
0

Relevant Questions

asked 2021-04-11
The equation F=−vex(dm/dt) for the thrust on a rocket, can also be applied to an airplane propeller. In fact, there are two contributions to the thrust: one positive and one negative. The positive contribution comes from air pushed backward, away from the propeller (so dm/dt<0), at a speed vex relative to the propeller. The negative contribution comes from this same quantity of air flowing into the front of the propeller (so dm/dt>0) at speed v, equal to the speed of the airplane through the air.
For a Cessna 182 (a single-engine airplane) flying at 130 km/h, 150 kg of air flows through the propeller each second and the propeller develops a net thrust of 1300 N. Determine the speed increase (in km/h) that the propeller imparts to the air.
asked 2021-02-25
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
asked 2021-04-16
A child is playing on the floor of a recreational vehicle (RV) asit moves along the highway at a constant velocity. He has atoy cannon, which shoots a marble at a fixed angle and speed withrespect to the floor. The cannon can be aimed toward thefront or the rear of the RV. Is the range toward the frontthe same as, less than, or greater than the range toward the rear?Answer this question (a) from the child's point of view and (b)from the point of view of an observer standing still on the ground.Justify your answers.
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors \(\displaystyle\hat{{{i}}}\) and \(\displaystyle\hat{{{j}}}\).
1. (Enter in box 1) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+\)
5. ( Enter in box 5) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
asked 2020-12-15
Expanding isosceles triangle The legs of an isosceles right tri- angle increase in length at a rate of 2 m/s.
a. At what rate is the area of the triangle changing when the legs are 2 m long?
b. At what rate is the area of the triangle changing when the hypot- enuse is 1 m long?
c. At what rate is the length of the hypotenuse changing?
asked 2021-05-12
An electron is fired at a speed of \(\displaystyle{v}_{{0}}={5.6}\times{10}^{{6}}\) m/s and at an angle of \(\displaystyle\theta_{{0}}=–{45}^{\circ}\) between two parallel conductingplates that are D=2.0 mm apart, as in Figure. Ifthe potential difference between the plates is \(\displaystyle\triangle{V}={100}\ {V}\), determine (a) how close d the electron will get to the bottom plate and (b) where the electron will strike the top plate.
asked 2021-03-15
Three long wires (wire 1, wire 2,and wire 3) are coplanar and hang vertically. The distance betweenwire 1 and wire 2 is 16.0 cm. On theleft, wire 1 carries an upward current of 1.50 A. To the right,wire 2 carries a downward current of 3.40 A. Wire 3 is located such that when itcarries a certain current, no net force acts upon any of the wires.
(a) Find the position of wire 3, relative to wire 1.
(b) Find the magnitude and direction of the current in wire 3.
asked 2020-11-05
Expanding isosceles triangle The legs of an isosceles right tri- angle increase in length at a rate of 2 m/s.
a. At what rate is the area of the triangle changing when the legs are 2 m long?
b. At what rate is the area of the triangle changing when the hypot- enuse is 1 m long?
c. At what rate is the length of the hypotenuse changing?
asked 2020-12-13
Expanding isosceles triangle The legs of an isosceles right tri- angle increase in length at a rate of 2 m/s.
a. At what rate is the area of the triangle changing when the legs are 2 m long?
b. At what rate is the area of the triangle changing when the hypot- enuse is 1 m long?
c. At what rate is the length of the hypotenuse changing?
...