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A roller coaster, starting at a height of 324 feet, goes down a hill and through an
underground tunnel. The function h(t) = t³-41²-81t + 324 represents the
coaster's height h (in feet) after t seconds, where. 0 ≤ t ≤ 10. How long is the
coaster in the tunnel?

2 Answers

2 votes

did you mean the function is

h(t) = t³ - 4t² - 81t + 324

the rest of this answer bases on this.

the roller coaster enters the underground tunnel at 0 ft height, and leaves the tunnel again at 0 ft height.

so, we need to find the zeros of the given function.

the difference between the zeros is the time the coaster is in the tunnel.

h(t) = 0 = t³ - 4t² - 81t + 324

a polynomial of the 3rd degree has actually 3 zeros.

we try to do factorization, which identified the zeros : every factor has a value for t to turn 0 and bring therefore the whole multiplication result to 0.

so, we need 3 factors :

(t + a)(t + b)(t + c) = t³ - 4t² - 81t + 324

(t² + (a+b)t + ab)(t + c)

t³ + ct² + (a+b)t² + c(a+b)t + abt + abc

t³ + (a+b+c)t² + (ca+cb+ab)t + abc

t³ = t³ done

a + b + c = -4

ca + cb + ab = -81

abc = 324

now, we have to solve this.

out of

a + b + c = -4

we know that at least one of the 3 variables has to be negative.

out of

abc = 324

we know then that exactly 2 of them have to be negative and one positive.

the factors of 324 are

1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, and 324

the prime factors are

2, 2, 3, 3, 3, 3

when combining them they create all the factors.

we are looking for a combination of 3 factors.

and they must be relatively close to each other, so that their sum can be so small as -4.

with a little trial and error we find that

324 = 4 × 9 × 9

for example 3×3×36 or 2×2×81 represent factors with sizes that are too far apart to provide a sum of -4 with whatever signs we can assign to them.

4, 9, 9 can be added to -4 only if 4 is negative and one 9 is negative.

324 = -4×9×-9

so, (a, b, c) for our factorization is

(-9, -4, 9)

and the factorization is then

(t - 9)(t - 4)(t + 9) = t³ - 4t² - 81t + 324

and the zeros are correspondingly

t = 9, 4, -9

t = -9 does not make any sense in our context (there cannot be negative time, and it is outside the given interval of 0 <= t <= 10).

so, what is left as regular solutions : t = 9, 4

that means the coaster enters the tunnel after 4 seconds and leaves it again after 9 seconds.

therefore,

the coaster is in the tunnel for 9-4 = 5 seconds.

User Tynn
by
7.4k points
7 votes

Final answer:

The roller coaster is in the tunnel for 3 seconds.

Step-by-step explanation:

The given function that represents the height of the roller coaster is h(t) = t³-41²-81t + 324. The coaster is in the tunnel when its height is less than or equal to 0, since the tunnel is at ground level. So we need to find the value of t when h(t) <= 0.

To solve this, we can set h(t) = 0 and solve for t:

t³-41²-81t + 324 = 0

Using factoring or the quadratic formula, we find two values of t that make the equation true: t = 3 and t = 6. Therefore, the roller coaster is in the tunnel for a total of 6 - 3 = 3 seconds.

User Tbleher
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7.7k points