A geometric pattern is formed by inscribing squares within squares. The largest square has side length 8 units. Each subsequent square is formed by connecting the midpoints of the sides of the previous square. As this process continues infinitely, what is the limit of the sum of the areas of all the squares?Answer: ______________
Tane is a marine biologist studying the population growth of a rare species of fish in a protected lagoon. The population size, in hundreds, is modeled by the function P(t) = (t^3 - 27)/(t - 3) for t ≠ 3, where t represents time in years since the study began. Due to a data collection gap, the population at exactly t = 3 years was not recorded. Tane needs to determine what population value the function approaches as time gets arbitrarily close to 3 years to complete his report. What is the limit of P(t) as t approaches 3?Answer: ______________
Liam is designing a roller coaster and needs to analyze the smoothness of a transition between two sections of track. The height of the track is modeled by the function f(x) = (x² - 4)/(x - 2) for x ≠ 2. To ensure a smooth ride, Liam needs to determine if the track height approaches a specific value as the roller coaster approaches x = 2. What value does the height approach as x approaches 2?Answer: ______________
Step 1: Direct substitution gives (16 - 16)/(2 - 2) = 0/0, an indeterminate form.
Step 2: Multiply numerator and denominator by the conjugate of the denominator, (√x + 2):
lim_{x→4} (x² - 16)/(√x - 2) * (√x + 2)/(√x + 2) = lim_{x→4} ((x² - 16)(√x + 2))/((√x - 2)(√x + 2))
Step 3: Simplify the denominator: (√x - 2)(√x + 2) = x - 4.
Step 4: Factor the numerator: x² - 16 = (x - 4)(x + 4).
So the expression becomes: lim_{x→4} ((x - 4)(x + 4)(√x + 2))/(x - 4).
Step 5: Cancel the common factor (x - 4) (since x ≠ 4 in the limit):
= lim_{x→4} (x + 4)(√x + 2).
Step 6: Substitute x = 4: (4 + 4)(√4 + 2) = (8)(2 + 2) = 8 * 4 = 32.
The answer is 32.
A geometric pattern is formed by inscribing squares within squares. The largest square has side length 8 units. Each subsequent square is formed by connecting the midpoints of the sides of the previous square. As this process continues infinitely, what is the limit of the sum of the areas of all the squares?Answer: 128 Solution: We start with a square of side length \( 8 \). Area of first square = \( 8^2 = 64 \). Each smaller square is formed by connecting the midpoints of the previous square's sides.Full step-by-step solution
Let's go step-by-step.
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**Step 1: Understanding the problem**
We start with a square of side length \( 8 \).
Area of first square = \( 8^2 = 64 \).
Each smaller square is formed by connecting the midpoints of the previous square's sides.
This smaller square is rotated 45 degrees and has side length equal to \( \frac{s}{\sqrt{2}} \) of the previous square's side length \( s \).
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**Step 2: Side length sequence**
First square: side \( a_1 = 8 \).
Second square: side \( a_2 = 8 / \sqrt{2} \).
Third square: side \( a_3 = (8 / \sqrt{2}) / \sqrt{2} = 8 / (\sqrt{2})^2 = 8 / 2 \).
Fourth square: side \( a_4 = (8 / 2) / \sqrt{2} = 8 / (2\sqrt{2}) \), etc.
General formula:
\( a_n = \frac{8}{(\sqrt{2})^{n-1}} \).
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**Step 3: Area sequence**
Area of nth square = \( (a_n)^2 \).
\( A_1 = 64 \).
\( A_2 = \left( \frac{8}{\sqrt{2}} \right)^2 = \frac{64}{2} = 32 \).
\( A_3 = \left( \frac{8}{2} \right)^2 = 16 \).
\( A_4 = \left( \frac{8}{2\sqrt{2}} \right)^2 = \frac{64}{8} = 8 \).
\( A_5 = \left( \frac{8}{4} \right)^2 = 4 \).
So the areas are:
64, 32, 16, 8, 4, 2, 1, 0.5, …
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**Step 4: Sum of areas**
We have an infinite geometric series:
\( S = 64 + 32 + 16 + 8 + 4 + 2 + 1 + 0.5 + \dots \)
First term \( A = 64 \), common ratio \( r = 1/2 \).
Sum of infinite geometric series = \( \frac{A}{1 - r} \).
\( S = \frac{64}{1 - 1/2} = \frac{64}{1/2} = 64 \times 2 = 128 \).
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**Step 5: Conclusion**
The limit of the sum of the areas of all the squares is \( 128 \).
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**Final answer:** 128
lim_(x→∞) (5x³ + 2x² - 7)/(3x³ - x + 4) = ?Answer: 5/3 Solution: Identify the degrees of numerator and denominator. Both are degree 3. For rational functions where degrees are equal, divide the leading coefficients.Full step-by-step solution
Step 1: Identify the degrees of numerator and denominator. Both are degree 3.
Step 2: For rational functions where degrees are equal, divide the leading coefficients.
Step 3: Leading coefficient of numerator is 5, leading coefficient of denominator is 3.
Step 4: The limit equals 5/3.
The answer is 5/3.
Tane is a marine biologist studying the population growth of a rare species of fish in a protected lagoon. The population size, in hundreds, is modeled by the function P(t) = (t^3 - 27)/(t - 3) for t ≠ 3, where t represents time in years since the study began. Due to a data collection gap, the population at exactly t = 3 years was not recorded. Tane needs to determine what population value the function approaches as time gets arbitrarily close to 3 years to complete his report. What is the limit of P(t) as t approaches 3?Answer: 27 Solution: Write the limit: lim(t→3) (t^3 - 27)/(t - 3) Factor the numerator using difference of cubes: t^3 - 27 = (t - 3)(t^2 + 3t + 9) Rewrite the function: P(t) = [(t - 3)(t^2 + 3t + 9)]/(t - 3) Cancel the common factor (t - 3): P(t) = t^2 + 3t + 9 for t ≠ 3 Evaluate the simplified function at t = 3:…Full step-by-step solution
Step 1: Write the limit: lim(t→3) (t^3 - 27)/(t - 3)
Step 2: Factor the numerator using difference of cubes: t^3 - 27 = (t - 3)(t^2 + 3t + 9)
Step 3: Rewrite the function: P(t) = [(t - 3)(t^2 + 3t + 9)]/(t - 3)
Step 4: Cancel the common factor (t - 3): P(t) = t^2 + 3t + 9 for t ≠ 3
Step 5: Evaluate the simplified function at t = 3: 3^2 + 3(3) + 9 = 9 + 9 + 9 = 27
The answer is 27.
Liam is designing a roller coaster and needs to analyze the smoothness of a transition between two sections of track. The height of the track is modeled by the function f(x) = (x² - 4)/(x - 2) for x ≠ 2. To ensure a smooth ride, Liam needs to determine if the track height approaches a specific value as the roller coaster approaches x = 2. What value does the height approach as x approaches 2?Answer: 4 Solution: f(x) = (x² - 4)/(x - 2) for x ≠ 2. We want the value f(x) approaches as x approaches 2. The numerator is x² - 4, which is a difference of squares: x² - 4 = (x - 2)(x + 2).Full step-by-step solution
Let's solve step by step.
We are given:
f(x) = (x² - 4)/(x - 2) for x ≠ 2.
We want the value f(x) approaches as x approaches 2.
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**Step 1: Factor the numerator**
The numerator is x² - 4, which is a difference of squares:
x² - 4 = (x - 2)(x + 2).
So f(x) = (x - 2)(x + 2) / (x - 2) for x ≠ 2.
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**Step 2: Simplify**
Since x ≠ 2, we can cancel (x - 2) from numerator and denominator:
f(x) = x + 2 for x ≠ 2.
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**Step 3: Take the limit as x approaches 2**
As x → 2, f(x) = x + 2 → 2 + 2 = 4.
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**Step 4: Conclusion**
Even though f(x) is not defined at x = 2, the limit exists and equals 4.
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**Final answer:** 4
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