1 angle of a rhombus measures 110°, and the shorter diagonal is 4 inches long. what is the length of the side?
4sqrt2
4
3
3.5
An auto weighing 2,000 pounds is on a street inclined at 10° with the horizontal. Find the force necessary to prevent the car from rolling down the hill.
347 pounds
14,397 pounds
2,462 pounds
The Cartesian equation, y = 4, is equivalent to the polar equation, r^2 = 4 sin θ.
True
False
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These are all trigonometry-related.
1. Draw it!
A rhombus is equilateral, and its angles are alternately supplementary -- each adjacent pair of them sum to 180º. So of the two different angles in this rhombus, 110º is the larger one.
And that makes the diagonal from that vertex, the shorter one.
So that 4-inch diagonal is the base of an isosceles triangle, with base angles both equal to:
θ = ½· 110º = 55º
Draw the altitude of that isosceles triangle, and isolate either of the two congruent right triangles that will make. Its leg, that was half of the base of the isosceles triangle, is 2 inches, the angle adjacent to it is θ, so the hypotenuse, which is one side of the rhombus, is
s = 2 in / cos(55º) ≈ 3.487 in
2. Apply a little sanity/common sense. If a heavy, rollable object is on a level surface, it requires no force to prevent it from rolling. If you incline the surface just a teeny bit, the force now required, is also pretty small; and increases the more you incline the surface, until that surface becomes vertical, when it will require a force equal to the entire weight of the object to hold it in place.
So it will never need more force than that, no matter what the inclination angle.
You've just ruled out two of the three answers.
If you want to know why the remaining one is correct, draw a right triangle and consider forces and components.
3. Draw it!
Plot y=4, then draw a radial line from the origin to some point (x,4) on the line.
Label the length of the radial line, r, and the angle from the +x-axis, counterclockwise to the radial line, θ.
Now draw the line segment from (x,0) to (x,4). You now have a right triangle. Express r in terms of θ (without using x!), and that's your equation, in polar coordinates, for the line, y=4, because it applies to any point (x,4) you would have chosen at the start.
Or you could go back to the sanity approach. In the equation
r² = 4 sinθ
• What's the biggest that sinθ can be?
• What's that make the biggest that r² can be?
• And what's that make the biggest value r can have?
• Are there any points on the line, y=4, that are farther than that from the origin?
• • Like, ALL of them?