Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Vlad is playing on a swing set. His horizontal distance D(t) (in m ) from the center (where being behind the center means a negative distance) as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*cos(b*t)+d. At t=0, when he pushes off, he is 1m behind the center, which is as far back as he goes. The swing reaches the center (pi)/(6) seconds later. Find D(t). t should be in radians. D(t)=◻

Vlad is playing on a swing set.\newlineHis horizontal distance D(t) D(t) (in m) from the center (where being behind the center means a negative distance) as a function of time t t (in seconds) can be modeled by a sinusoidal expression of the form a⋅cos⁡(b⋅t)+d a \cdot \cos (b \cdot t)+d .\newlineAt t=0 t=0 , when he pushes off, he is 1 m 1 \mathrm{~m} behind the center, which is as far back as he goes. The swing reaches the center π6 \frac{\pi}{6} seconds later.\newlineFind D(t) D(t) .\newlinet t should be in radians.\newlineD(t)=□ D(t)=\square

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Interpret parts of quadratic expressions: word problemsright chevron icon

Full solution

Q. Vlad is playing on a swing set.\newlineHis horizontal distance D(t) D(t) (in m) from the center (where being behind the center means a negative distance) as a function of time t t (in seconds) can be modeled by a sinusoidal expression of the form a⋅cos⁡(b⋅t)+d a \cdot \cos (b \cdot t)+d .\newlineAt t=0 t=0 , when he pushes off, he is 1 m 1 \mathrm{~m} behind the center, which is as far back as he goes. The swing reaches the center π6 \frac{\pi}{6} seconds later.\newlineFind D(t) D(t) .\newlinet t should be in radians.\newlineD(t)=□ D(t)=\square
  1. Identify Amplitude: Identify the amplitude of the sinusoidal function.\newlineThe amplitude aa is the maximum distance from the center, which is given as 1 m1\,\text{m} behind the center at t=0t=0.\newlineTherefore, a=1a = 1.
  2. Determine Horizontal Shift: Determine the horizontal shift dd of the sinusoidal function.\newlineSince Vlad starts 11m behind the center and that is the farthest point, the horizontal shift dd is 00.
  3. Determine Period: Determine the period of the sinusoidal function.\newlineWe know that the swing reaches the center (π/6)(\pi/6) seconds later, which is a quarter of the period of a cosine function. To find the full period TT, we multiply this time by 44.\newlineT=(π/6)×4=(2π/3)T = (\pi/6) \times 4 = (2\pi/3) seconds.
  4. Calculate Value of b: Calculate the value of b, which is related to the period of the sinusoidal function. The period TT is related to bb by the formula T=2πbT = \frac{2\pi}{b}. We have T=2π3T = \frac{2\pi}{3}, so we can solve for bb: 2π3=2πb\frac{2\pi}{3} = \frac{2\pi}{b} Cross-multiply to solve for bb: b⋅2π3=2πb \cdot \frac{2\pi}{3} = 2\pi b=2π2π3b = \frac{2\pi}{\frac{2\pi}{3}} b=3b = 3.
  5. Write Sinusoidal Function: Write the sinusoidal function for D(t)D(t). We have a=1a = 1, b=3b = 3, and d=0d = 0. Since Vlad starts 1m1m behind the center, we use a cosine function that starts at its maximum value. The function is: D(t)=acos⁡(b⋅t)+dD(t) = a\cos(b\cdot t) + d D(t)=1cos⁡(3⋅t)+0D(t) = 1\cos(3\cdot t) + 0 D(t)=cos⁡(3⋅t)D(t) = \cos(3\cdot t).

More problems from Interpret parts of quadratic expressions: word problems

Question
Solve by completing the square.\newlinem2−10m−29=0m^2 - 10m - 29 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newline`m` = ____ or `m` = _____
Get tutor helpright-arrow

Posted 8 months ago

Question
Find g(x) g(x) , where g(x) g(x) is the translation 5 5 units up of f(x)=x2 f(x)=x^2 .\newlineWrite your answer in the form a(x–h)2+k a(x–h)^2+k , where a a , h h , and k k are integers.\newlineg(x)= g(x)= ____
Get tutor helpright-arrow

Posted 7 months ago

Question
What is the range of this quadratic function?\newliney=x2−4x+4y = x^2 - 4x + 4\newlineChoices:\newline{y∣y≥2}\left\{y \mid y \geq 2\right\}\newline{y∣y≤0}\left\{y \mid y \leq 0\right\}\newline{y∣y≥0}\left\{y \mid y \geq 0\right\}\newlineall real numbers\text{all real numbers}
Get tutor helpright-arrow

Posted 5 months ago

Question
Write the equation of the parabola that passes through the points (1,0)(1,0), (2,0)(2,0), and (3,–16)(3,\text{–}16). Write your answer in the form y=a(x–p)(x–q)y = a(x – p)(x – q), where aa, pp, and qq are integers, decimals, or simplified fractions.\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.\newlinef2+8f+_____f^2 + 8f + \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Solve for h h .\newlineh2+39h=0 h^2 + 39h = 0 \newline\newlineWrite each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.\newlineh= h = ____\newline
Get tutor helpright-arrow

Posted 5 months ago

Question
Write a quadratic function with zeros −9-9 and −7-7.\newlineWrite your answer using the variable xx and in standard form with a leading coefficient of 11.\newlinef(x)=_____f(x) = \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Find the equation of the axis of symmetry for the parabola y=x2y = x^2. \newlineSimplify any numbers and write them as proper fractions, improper fractions, or integers.\newline‾\underline{\hspace{3cm}}
Get tutor helpright-arrow

Posted 4 months ago

Question
Find g(x)g(x), where g(x)g(x) is the translation 88 units up of f(x)=x2f(x) = x^2.\newlineWrite your answer in the form a(x–h)2+ka(x – h)^2 + k, where aa, hh, and kk are integers.\newlineg(x)=g(x) = ______\newline
Get tutor helpright-arrow

Posted 4 months ago

Question
Solve for xx.\newlinex2=1x^2 = 1\newline\newlineWrite your answer in simplified, rationalized form.\newlinex=x = ______ or x=x = ______\newline
Get tutor helpright-arrow

Posted 5 months ago

View all (97)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant