Choose Yes or No to tell whether there is more than one way to draw a triangle with the given measures. AB = 15 cm, BC = 20 cm, ∠B = 40° AB = 20 cm, AC = 15 cm, ∠C = 40° BC = 10 cm, AB = 5 cm, ∠A = 20° AC = 5 cm, BC = 10 cm, AB = 14 cm

Question:

a. AB = 15 cm, BC = 20 cm, ∠B = 40°

b. AB = 20 cm, AC = 15 cm, ∠C = 40°

c. BC = 10 cm, AB = 5 cm, ∠A = 20°

d. AC = 5 cm, BC = 10 cm, AB = 14 cm

The correct options are;

a. No

b. Yes

c. Yes

d. No

Step-by-step explanation:

a. AB = 15 cm, BC = 20 cm, ∠B = 40°

No

The above dimensions include two sides and an included angle, therefore, the triangle is well defined, such that the dimensions of the third side can be found by the given information using cosine rule

b. AB = 20 cm, AC = 15 cm, ∠C = 40°

Yes,

The above dimensions include two sides and a non included angle, therefore, the triangle is not well defined, such that the third side and the other two angles can be made to to make up to a different triangular configuration can be made

c. BC = 10 cm, AB = 5 cm, ∠A = 20°

Yes

The above triangle could be made into one of two triangles

d. AC = 5 cm, BC = 10 cm, AB = 14 cm

No

The above triangle is well defined with all side dimensions known and the angles can be found by cosine rule.

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

i would say true? if the unit was informational and it "refreshed your understanding", then true.