How do I solve these problems (Algebra Help)?

Question

This may seem like a lot, but I'm really sorry. I've moved schools so much this year that I've missed out on a lot of Algebra units, since each class had a different pace. Now its time for the end of course test and I'm afraid of failing. My teacher now is really mean and I'm scared to ask her questions. How would I solve the following problems? 

(1) Shawn and Curtis went for a walk. Shawn began walking 20 seconds earlier than Curtis at a speed of 5 feet per second. Curtis walked at a speed of 6 feet per second. For how many seconds had Shawn been walking at the moment when the two boys had walked exactly the same distance? 

(2) What is the smallest of 3 consecutive positive integers if the product of the smaller two integers is 5 less than 5 time the largest integer? 

(3) The function f(t)= -5t(to the second power) + 20t + 60 models the approximate height of an object "t" seconds after it is launched. How many seconds does it take the object to hit the ground? 

(4) The larger leg of a right triangle is 3cm longer than its smaller leg. The hypotenuse is 6cm longer than the smaller leg. How many centimeters long is the smaller leg? 

I have the answers to these b/c they are in the back of our study book (to check ourselves), but for the course test, I would really like to know how to actually solve these. All answers are appreciated! Thank you for the help.

IIl · Accepted Answer

1.
t = Curtis's time
t + 20 = Shawn's time; he walked 20 seconds more than Curtis.

distance = rate • time

Curtis's distance = Curtis's rate • t
                                 = 6t

Shawn's distance = Shawn's rate • (t + 20)
                                 = 5(t + 20)
                                 = 5t + 100

The moment they traveled the same distance, Curtis's distance = Shawn's distance.
6t = 5t + 100
6t − 5t = 100
t = 100

This means Curtis has walked for t = 100 seconds and Shawn walked for
t + 20 = 100 + 20 = 120 seconds.

2. Let x be the first (smallest) positive integer. Then the next consecutive integer is
x + 1 and the third consecutive integer is x + 2. Thus, x < x + 1 < x + 2.

The product of the smaller two integers is: x(x + 1)
5 less than 5 times the largest integer is: 5(x + 2) − 5

x(x + 1) = 5(x + 2) − 5
x² + x = 5x + 10 − 5
x² + x = 5x + 5
x² + x − 5x − 5 = 0
x² − 4x − 5 = 0
(x + 1)(x − 5) = 0

x + 1 = 0      x − 5 = 0
       x = −1            x = 5

There are two solutions. It can't be x = −1 because the problem states that it's a positive integer, so x = 5 is the answer.

Check: The consecutive integers are 5, 6, and 7.
(5)(6) = 5(7) − 5
     30 = 30

3. f(t) = −5t² + 20t + 60

Here, f(t) is the height (the numbers on the y-axis). When the object hits the ground its height is zero, and the equation becomes 0 = −5t² + 20t + 60.

0 = −5t² + 20t + 60
0 = −5(t² − 4t − 12)
0 = −5(t − 6)(t + 2)

t − 6 = 0      t + 2 = 0
       t = 6            t = −2

You can't have a negative time so t = 6 seconds is the answer. If you're not great at factoring, you can use the quadratic formula instead. Might help on an exam.

Check: f(6) = −5(6)² + 20(6) + 60 = 0

4. x = smaller leg
x + 3 = larger leg
x + 6 = hypotenuse

By the Pythagorean Theorem
a² + b² = c²
x² + (x + 3)² = (x + 6)²
x² + (x + 3)(x + 3) = (x + 6)(x + 6)
x² + (x² + 3x + 3x + 9) = x² + 6x + 6x + 36
2x² + 6x + 9 = x² + 12x + 36
2x² − x² + 6x − 12x + 9 − 36 = 0
x² − 6x − 27 = 0
(x − 9)(x + 3) = 0

x − 9 = 0      x + 3 = 0
       x = 9            x = −3

You can't have a negative leg, so x = 9 is the answer.

Check:
x = 9
x + 3 = 12
x + 6 = 15

9² + 12² = 15²
        225 = 225

Anonymous · Answer

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? · Answer

So why did you not provide the answers, so that we can check our work, and not mislead you???

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How do I solve these problems (Algebra Help)?

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