P is inversely proportional to q^3, when p has the value 27, q has the value 3. i) find the formula which expresses p in terms of q. P= r is directly proportional to q, when r has the value 10, q has the value 2. ii) prove that r=kp^n and so find the value of k and n.

Step-by-step explanation:

i) If p is inversely proportional to q³, this is expressed as p ∝ q³

p = k/q³ where q is the proportionality constant.

Given p = 27 and q = 3

Substitute this values into the formula

27 = k/(3)³

27 = k/27

k = 27×27

k = 729

Substitute k = 1 into the formula

p = 729/(q)³

Hence the formula which expresses p in terms of q is p = 729/q³

ii) If r is directly proportional to q, it is expressed as r ∝ q

r = kq

If r = 10 and q = 2

Substitute

10 = 2k

k = 10/2

k = 5

The equation becomes

r = 5q... 1

From 1) p = 729/q³

q³ = 729/p

q = 9/³√p

q = 9/p^1/32

Substitute equation 2 into 1

r = 5(9/p^1/3)

r = 45p^-1/3

Compare r = 45p^-1/3 with r = kp^n

k = 45 and n = -1/3

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step-by-step explanation:

1. 180°. the sum of the interior angles of a triangle is 180°, for the equilateral triangle that it has three equals side is 60°+60°+60°=180°.

2. 60°. each interior angle of a equilateral triangle is 60°

3. 120°. the measure of ∠m is 120°, the sum of the angles ∠m and ∠n is equal to 180°. so, in order to find the measure of ∠m, we have ∠m+∠n=180° > ∠m=180°-∠n > ∠m=180°-60°=120°.

4. 360°. the sum of the exterior angles of an equilateral triangle is ∠m+∠r+∠x=120°+120°+120°=360°. the exterior angles of any triangle always add 360°.