I need answers for both questions thank you

Some math problems require an exact answer, while for others, an approximate answer is good enough.

When a problem involves measurement of a real-world quantity, there is always some level of approximation happening. For example, consider two different problems involving measures of time. If the problem involves sprinters running the 100100 -meter dash, you will need to use precise measurements: hundredths of a second or better. If the problem involves people’s ages, it’s usually enough to approximate to the nearest year.

Sometimes when a problem involves square roots or other irrational numbers like pi , or even a complicated fraction, it’s useful to use decimal approximations—at least at the end, when you’re reporting your answer.

Example 1:

Deanna is making a wire fence for her garden in the shape below. Find the length of the hypotenuse of the triangle in meters.



Using the Pythagorean Theorem, we get:

x2=32+72x2=9+49x=56−−√x2=32+72x2=9+49x=56

So the hypotenuse is 56−−√56 meters long. That’s the exact answer. But having the answer in this format isn’t very useful if we’re trying to build something. How do you cut a piece of wire 56−−√56 m long?

A good calculator will tell you the value of 56−−√56 correct to 3030 decimal places:

56−−√≈7.48331477354788277116749746463356≈7.483314773547882771167497464633

Even this is an approximation. But it’s a much better approximation that you need. In this case, rounding the value to the nearest centimeter (hundredth of a meter) is probably enough.

56−−√=7.4856=7.48

The accuracy of a measurement or approximation is the degree of closeness to the exact value. The error is the difference between the approximation and the exact value.

When you’re working on multi-step problems, you have to be careful with approximations. Sometimes, an error that is acceptable at one step can get multiplied into a larger error by the end.

Example 2:

A plastic disk is the shape of a circle exactly 1111 inches in diameter. Find the combined area of 10,00010,000 such disks.

Suppose you use 3.143.14 as an approximation for ππ . Using the formula for the area of a circle, we get the area of one disk as:

A=πr2=π⋅112=121π≈121(3.14)=379.94A=πr2=π⋅112=121π≈121(3.14)=379.94

Multiply this value by 10,00010,000 to get the combined area of 10,00010,000 disks.

379.94×10,000=3,799,400379.94×10,000=3,799,400

This gives an answer of 3,799,4003,799,400 square inches. But wait!

See what happens when we use a more accurate value for ππ like 3.14163.1416 :