Please help would really appreciate it

Mathematics

1 answer:

laiz [17]3 years ago

6 0

The given data is angle ∡RSU and angle ∡UST are complementary and

angle ∡RSU= 8k-1 and angle ∡UST = 7 (k-2)

The sum of the complementary angles is 90°

∡RSU + ∡UST = 90° => 8k-1 + 7 (k-2) = 90 => 8k-1+7k-14=90 => 15k - 15 = 90 => 15k = 90+15 => 15k = 107 => k= 105/15 = 7° then

∡RSU= 8k-1= 8*7-1= 55°

∡UST= 7 (k-2)= 7 (7-2)= 7*5= 35°

The correct answer is ∡RSU= 55° and ∡UST= 35°

Good luck!!!

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For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? Vertical?

Bond [772]

Given that $r=\theta$ , then $r'=1$

The slope of a tangent line in the polar coordinate is given by:

$m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$

Thus, we have:

$m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta}$

Part A:

For horizontal tangent lines, m = 0.

Thus, we have:

$\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488

Part B:

For vertical tangent lines, $\frac{1}{m} =0$

Thus, we have:

$\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713

3 0

3 years ago

SOMEONE HELP PLEASE! ASAP!

Luba_88 [7]

When two lines intersect, the angles across from the intersection (the "X") are congruent (equal), and referred to as "vertical angles."
So by the rule of vertical angles are congruent, we can set them equal:
${x}^{2} - 6x = 1\div2 \: x + 42 \\ {x}^{2} - 6x - 1 \div 2 \: x = 42 \\ {x}^{2} - 6.5x = 42$
${x}^{2} - 6.5x - 42 = 0 \\ multiply \: both\: sides \: by \: 2 \: to \: get \\ rid \: of \: the \: decimal \\ 2({x}^{2} - 6.5x - 42) = 2(0)$
$2 {x}^{2} - 13x - 84 = 0$
to factor we need the factors of 84:
1×84, 2×42, 3×28, 4×21, 6×14
Now we need one of those factors ×2 ( from the x^2 term) minus the other factor to equal 13 (the middle term)
it turns out that if we use 4×21: 4×2 = 8
and 21-8 = 13. Woohoo! we got it
So now we have:
$2 {x}^{2} - 13x - 84 = 0 \\ (2x - 21)(x + 4) = 0$
Now set each factor = 0
1) 2x - 21 = 0 and 2) x + 4 = 0
1) 2x - 21 = 0, 2x = 21, x = 21/2 = 10.5
2) x + 4 = 0, x = -4

Now we plug in each answer to check:
1) x = 10.5
${x}^{2} - 6x = {(10.5)}^{2} - 6(10.5) \\ = 110.25 - 63 = 47.25$
1/2x + 42 = 1/2(10.5) + 42 = 5.25 + 42 = 47.25
47.25 = 47.25
BAM SO IT LOOKS LIKE THAT'S IT!!
But let's check 2) first:
2) x = -4
${( - 4)}^{2} - 6( - 4) = 16 + 24 = 40$
1/2x + 42 = 1/2(-4) + 42 = -2 + 42 = 40
40 = 40, so both work!!
But it asks for m<1, and <1 is supplementary because it lies on same line, meaning the sum of both = 189
Therefore m<1 = 180 - 47.25 = 132.75°
OR m<1 = 180 - 40 = 140°

6 0

3 years ago

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Llana [10]

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3 0

3 years ago

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zmey [24]

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Step-by-step explanation: