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1. A recipe calls for 45 mL of wine. How many tablespoons of wine should you use?

2. You are making Maple Pecan Pies. Each crust will require 10 ounces of flour. You will be making 25 pies. How many kilograms of flour should you use?

3. You purchase two cases of orange juice. Each case contains four 1-gallon containers of juice. You are serving the orange juice for a breakfast meeting. Each of the 160 guests receives a 6-fluid-ounce glass of juice. What percent of the juice was used?

4. You have 1.5 kilograms of sugar. You use 850 grams of the sugar. How many ounces of sugar do you have remaining?

5. You have 1¾ cups of water. How many milliliters of water do you have?

6. You have a 1-pound container of ground cloves. One tablespoon of ground cloves weighs ¼ ounce. How many cups of ground cloves are in the container of ground cloves? How many teaspoons of ground cloves are in the container?

7. You are making blueberry pies. Each pie requires 2 pints of blueberries. You are making 6 pies. If 1 cup of blueberries weighs 6 ounces, then how many pounds of blueberries should you use?

8. A recipe calls for 4 ounces of mustard seed. You use a half a cup. If mustard seed weighs 2/5 of an ounce per tablespoon, how much more or less in ounces of the mustard seed should you have used?

9. A cup of dried oregano weighs 3 ounces. In ounces, how much would 1 teaspoon of dried oregano weigh?

10. How many quarts of honey are in 2.5 kilograms of honey if 1 cup of honey weighs 12 ounces?

11. A recipe for Cranberry Bread calls for 11/3 cups of cranberries. If you are making 15 loaves and 1 cup of cranberries weighs 4 ounces, how many pounds of cranberries should you use?

Answer :

The radius to point P in polar coordinates is given by the generic form, [tex]r(ϕ)=d/cos(ϕ−α)[/tex]. The shortest distance between two points is a straight line using polar coordinates.

**a.** Consider a straight line that does not pass through the origin. Let the length of the perpendicular segment between the line and the origin be $d$, and let the polar angle made by the perpendicular segment be $\alpha$. Let $P$ be a point on the line, and let $\phi$ be the polar angle of $P$.

We can see that the radius to point $P$ in polar coordinates is given by the following equation:

[tex]r(\phi) = \frac{d}{\cos(\phi - \alpha)}[/tex]

This equation can be derived by considering the right triangle formed by the line, the perpendicular segment, and the radius to point $P$.

**b.** The shortest distance between two points in Euclidean space is a straight line. This can be shown using the Euler-Lagrange equations.

The Euler-Lagrange equations are a set of differential equations that can be used to find the extremum of a function subject to some constraint. In this case, the function we are trying to extremize is the distance between two points, and the constraint is that the distance must be a straight line.

The Euler-Lagrange equations for this problem can be written as follows:

[tex]\frac{d}{d\phi} \left[ \frac{d}{d\phi} \left[ \frac{1}{2} r^2 (\phi) \right] \right] = 0[/tex]

This equation can be solved to show that the shortest distance between two points is a straight line.

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