Answered

County X has a population density of 250 people per square mile. The total

population of the county is 150,000. Which geometric model could be the shape of

county X?

A a parallelogram with a base of 25 miles and a height of 25 miles

B a rectangle that is 15 miles long and 45 miles wide

C a right triangle with a leg that is 30 miles long and a hypotenuse that is

50 miles long

D a trapezoid with base lengths of 10 miles and 30 miles and a height of

25 miles

Verified Answer :

Cwith a 150,000 person population at 250 persons per square mile the country would be 600 square miles. The last leg would equal 40 miles and therefore making the area equal 600 square miles.

Answer:

Option C. is the correct option.

Step-by-step explanation:

County X has a population density of 250 people per square mile.

Total population of the county is 150000.

Total area which represents the population = \(\frac{\text{Total population}}{\text{Population density}}\) = \(\frac{150000}{250}=600\) square mile.

So geometric model which has the same area as 600 square miles will be the answer.

A). Area of a parallelogram = Base × height

= 25×25

= 625 square miles

which is not equal to 600 square miles so it's not the correct option.

B). Area of a rectangle = Length×width

= 15×45 = 675 square miles ≠ 600 square miles

Therefore, not the correct option.

C). Area of a right triangle = \(\frac{1}{2}(Base)(height)\)

Since length of hypotenuse is given so we will find the length of height.

Height = √[(Hypotenuse)²- (Base)²] = \(\sqrt{50^{2}-30^{2}}\)

= \(\sqrt{2500-900}\)

= \(\sqrt{1600}\)

= 40 miles

So area of the right triangle = \(\frac{1}{2}(40)(30)\)

= \(\frac{1200}{2}\)

= 600 square miles

Which matches the area of County X, so it's a correct option.

D). area of trapezoid = \(\frac{1}{2}[(base1 + base2)](height)\)

= \(\frac{1}{2}(10+30)(25)\)

= \(\frac{1}{2}(40)(25)\)

= 20×25

= 500 square miles ≠ 600 miles

Therefore, this option is not correct.

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