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2 years ago

We need to attach a support wire to Glendale's flagpole, which is 24 feet high. The

Mathematics

1 answer:

antoniya [11.8K]2 years ago

4 0

Step-by-step explanation:

this creates a right-angled triangle.

the right angle (90°) being the angle between flag pole and ground.

the angle ground-wire is 50°.

because we know that the sum of all angles in a triangle is always 180°, we know then that the angle flagpole-wire is

180 - 90 - 50 = 40°

so, we have the situation to know 1 side and all angles.

to get the other sides we use best the law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

where the sides and related angles are always opposite of each other.

in our case now we have

24/sin(50) = wire/sin(90) = wire/1 = wire = 31.32977494... ft

so, the rounded answer is that the wire must be

31.3 ft long

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6/7.

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3 years ago

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3 years ago

Suppose that r1 and r2 are roots of ar2 + br + c = 0 and that r1 = r2; then exp(r1t) and exp(r2t) are solutions of the different

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The Correct Question is:

Suppose that r1 and r2 are roots of ar² + br + c = 0 and that r1 = r2; then e^(r1t) and e^(r2t) are solutions of the differential equation

ay'' + by' + cy = 0.

Show that

φ (t; r1, r2) = [e^(r2t) - e^(r1t )]/(r2 - r1)

is a solution of the differential equation.

Answer:

φ (t; r1, r2) is a solution of the differential equation, and it shown.

Step-by-step explanation:

Given the differential equation

ay'' + by' + cy = 0

and r1 and r2 are the roots of its auxiliary equation.

We want to show that

φ (t; r1, r2) = [e^(r2t) - e^(r1t )]/(r2 - r1)

satisfies the given differential equation, that is

aφ'' + bφ' + cφ = 0 .....................(*)

Where φ = φ (t; r1, r2)

We now differentiate φ twice in succession, with respect to t.

φ' = [r2e^(r2t) - r1e^(r1t )]/(r2 - r1)

φ'' = [r2²e^(r2t) - r1²e^(r1t )]/(r2 - r1)

Using these in (*)

We have

a[r2e^(r2t) - r1e^(r1t )]/(r2 - r1) + [r2²e^(r2t) - r1²e^(r1t )]/(r2 - r1) + c[e^(r2t) - e^(r1t )]/(r2 - r1)

= [(ar2² + br2 + c)e^(r2t) - (ar1² + br1 + c)e^(r1t)]/(r1 - r2)

We know that r1 and r2 are the roots of the auxiliary equation

ar² + br + c = 0

and r1 = r2

This implies that

ar1² + br1 + c = ar2² + br2 + c = 0

And hence,

[(ar2² + br2 + c)e^(r2t) - (ar1² + br1 + c)e^(r1t)]/(r1 - r2) = 0

Therefore,

aφ'' + bφ' + cφ = 0

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