An amount of $2,500 is deposited in a savings account in a bank paying an annual interest rate of 4.2% compounded continuously. What will be the balance after 3 years?

The key words from the problem are "compounded continuously." That directly takes us to using the formula for continuous compound interest A=Pe^(rt). The key here is to recognize which letter of the formula corresponds to which value of the information provided. We know that t is time, which in our case is 3 years; we also know that the deposited amount is the principal P, therefore P=$2,500; and lastly, we know that r is the interest rate of 4.2%. First, we need to convert the percentage into a numerical value. We will do that by dividing 4.2 by 100. Therefore, r=4.2/100=.042. Now we can just plug in all the numbers in the formula, A=(2500)e^(.042*4). Since we are using the number e, with the help of a scientific calculator, we get A=2957.34.

In the right triangle ABC , m<ACB=90, m< ABC=60, and AC=4ft, what are the lengths of AB and BC?

Let's draw the triangle ABC and label the corresponding measures of the angles and the side AC. From the information above we know that <ABC=60. We also know that sin(ABC)=OPP/HYP (OPP=opposite side, HYP=hypotenuse). By looking at the drawing it is evident that the opposite side is AC and the hypotenuse is AB. Therefore, sin(ABC) = AC/AB. Let's multiply both sides by AB. Then we get AB(sin(ABC))=AC. Since we are looking for AB, now we divide both sides by sin(ABC) and we get AB=AC/(sin(ABC)). Then we substitute with the numerical information. We have AB=4/(sin(60)) which is equivalent to AB=4/sqrt(3)/2. After simplifying, we get AB=8/sqrt(3). After rationalizing, we get AB=8sqrt(3)/3. Now, all we have left is to find the measure of BC. Here we can apply the Pythagorean formula, which is AC^2+BC^2=AB^2. If we subtract AC^2 from both sides, we get BC^2=AB^2-AC^2. Then we can square root both sides to cancel the exponent of 2 on BC^2. And we get BC=sqrt(AB^2-AC^2). We substitute with the numerical information and get BC=sqrt( (8sqrt(3)/3)^2 - 4^2).

Prove that the triangle ABC is congruent to the triangle DEF. m<ABC=m<DEF, AB=DE, BC=EF.

There are 5 cases in which two triangles are congruent. We need to decide which of the cases will be applicable to our problem. If we draw the triangles, it will be obvious that we have two pairs of sides that are equal and a pair of equal angles that lie between those two sides. With that being said, the best approach would be to use the SAS (side, angle, side) case . By choosing it, we prove that ABC is congruent to DEF.