Common Core Standard HSF-LE.A.3 Questions

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Grade 11 Exponents CCSS: HSF-LE.A.3

The height of a sunflower is modeled by the linear function H(d) = 5d + 10. The height of a magic beanstalk is modeled by [math]M(d) = 0.1 (2)^d[/math]. Which plant will be taller?

The sunflower
The beanstalk
They will be the same height
It depends on the number of days d.

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

An environmental scientist is modeling the sequestration of carbon in two different forest management strategies. Strategy Alpha results in carbon storage modeled by the function Ca(y) = 50y² + 100y + 2000, where Ca is the total carbon in metric tons and y is the number of years. Strategy Beta results in carbon storage modeled by the function [math]C_beta(y) = 2000(1.07)^y[/math]. Which strategy will sequester more carbon in the long term?

The growth will eventually become equivalent.
There is not enough information to make a determination.
Strategy Alpha.
Strategy Beta.

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

A mechanical engineer is testing the wear and tear on two engine components. The wear on Component Z is modeled by [math]W_z(h) = 0.001h^2 + 0.1h[/math] (in mm of material lost). The wear on Component Y is modeled by [math]W_y(h) = 0.01(1.02)^h[/math], where h is operating hours. After a very long period of continuous operation (e.g., 10,000 hours), which component is predicted to have less wear?

Both components will have failed catastrophically.
Component Y will have less wear.
Component Z will have less wear.
The wear will be equivalent.

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

The table provides values for a quadratic function, a(x) = 5x², and an exponential function, b(x). Which statement is true?

b(x) > a(x) for x > 2. After the first intersection, the exponential function is always greater.
a(x) > b(x) for x > 3. The quadratic function's growth is faster than this particular exponential function's growth.
b(x) > a(x) for x ≥ 5. The functions intersect at x=2 and x=4. After the second intersection, the exponential function's growth becomes dominant.
The functions will intersect a third time, meaning a(x) will eventually become greater again for very large x.

Grade 11 Exponents CCSS: HSF-LE.A.3

A company offers two salary plans. Plan A: a flat $5,000 raise each year. Plan B: a 5% raise each year. After 30 years, which plan yields a higher salary?

Plan A
Plan B
They are equal
It depends on the starting salary

Grade 11 Exponents CCSS: HSF-LE.A.3

A civil engineer is analyzing the long-term structural deformation of two bridge designs under constant, heavy load. The deformation in Design A is modeled by the function [math]D_A(t) = 0.01t^2 + 0.5t + 2[/math] (in centimeters), where t is time in years. The deformation in Design B is modeled by [math]D_B(t) = 10(1.001)^t[/math]. After several thousand years, which design is predicted to show less deformation?

Both designs will have failed long before this timeframe.
The models are not valid for such extreme timescales.
Design A will exhibit less deformation.
Design B will exhibit less deformation.

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

The following table shows some of the values of functions [math]f(x)[/math], which is linear, and [math]g(x)[/math], which is exponential. What can be concluded from these values (and general properties of these functions), and why?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 2 [/math]	[math] 23 [/math]	[math] 4 [/math]
[math] 3 [/math]	[math] 33 [/math]	[math] 8 [/math]
[math] 4 [/math]	[math] 43 [/math]	[math] 16 [/math]
[math] 5 [/math]	[math] 53 [/math]	[math] 32 [/math]
[math] 6 [/math]	[math] 63 [/math]	[math] 64 [/math]
[math] 7 [/math]	[math] 73 [/math]	[math] 128 [/math]

[math]g(x) > f(x)[/math] for all values of [math]x[/math]. This is because [math]f(x)[/math] is an exponential function, which must always be greater than a linear function.
[math]g(x)>f(x)[/math] for [math]x>=6[/math]. Since the values of [math]g(x)[/math] continue to increase by ever greater amounts each unit interval, they will continue to be greater than the values of [math]f(x)[/math] which are only increasing by the same amount each unit interval.
[math]g(x) > f(x)[/math] for [math]6 <= x <= 45[/math]. If a linear function intersects an exponential function, it must do so twice. Therefore, the two functions will intersect again at approximately [math]x~~2^5.5 ~~ 45.3[/math], and thereafter [math]f(x) > g(x)[/math].
[math]g(x) > f(x)[/math] for [math]x>=6[/math] and [math]g(x) < f(x)[/math] for [math]x <= 5[/math]. Since [math]x~~5.5[/math] is the only point of intersection, and given the values in the table, [math]g(x)[/math] must be greater than [math]f(x)[/math] for values of [math]x[/math] greater than or equal to 6 and it must be less than [math]f(x)[/math] for values of [math]x[/math] less than or equal to 5.

Grade 11 Exponents CCSS: HSF-LE.A.3

The cost of a linear algorithm is C₁(n) = 100n. The cost of an exponential algorithm is C₂(n) = 2^n. For large input sizes n, which algorithm is more efficient?

The linear algorithm
The exponential algorithm
They are equally efficient
It depends on the value of n

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

The growth of two plants is modeled by a linear function f(x) and an exponential function g(x). What can be concluded?

g(x) > f(x) for x ≥ 3. The exponential plant's tripling growth causes it to surpass the linear plant that grows by 3 each week.
f(x) > g(x) for all x. The linear plant was taller initially and grows at a steady rate, so it will always be taller.
g(x) > f(x) for x ≥ 2. The values are equal between x=2 and x=3, and then g(x) becomes greater.
The functions will intersect again, meaning f(x) will eventually be taller for a very large x.

Grade 11 Exponents CCSS: HSF-LE.A.3

The number of users on a new social media platform increases exponentially, while the number of users on an established platform increases linearly. After several years, what is expected to happen?

The new platform's growth will slow down
The established platform will have more users
They will have the same number of users
The new platform will have more users

Grade 11 Exponents CCSS: HSF-LE.A.3

The distance traveled by a snail is [math]D_s(t) = 0.01t + 0.1[/math] (meters). The distance traveled by a bacteria is [math]D_b(t) = 0.0001 • (2)^t[/math] (meters). For large t, which travels farther?

The snail
The bacteria
They travel the same distance
It depends on t

Grade 11 Exponents CCSS: HSF-LE.A.3

The number of new social media followers for a celebrity is F(d) = 100d + 5000. For a viral video, it is [math]V(d) = 10 • (2)^d[/math]. After 10 days, which has more followers?

The celebrity
The viral video
They have the same number
It depends on the day

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

Analyze the values of the linear function h(x) and the exponential function k(x). What can be concluded?

k(x) > h(x) for all x > 1. Since k(x) is exponential, it is always greater than a linear function after they intersect.
k(x) > h(x) for x ≥ 3. The exponential function's doubling growth causes it to exceed and then dominate the linear function.
k(x) > h(x) for 3 ≤ x ≤ 20. The functions will intersect again, after which h(x) will be greater.
k(x) > h(x) for x ≥ 2 and k(x) < h(x) for x ≤ 1. The functions intersect only once at x=2, and the exponential grows faster thereafter.

Grade 11 Functions and Relations CCSS: HSF-LE.A.3

The following table gives some of the values of [math]f(x)[/math], a quadratic function, and [math]g(x)[/math], an exponential function. What can be concluded from these values (and general properties of the functions), and why?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 8 [/math]	[math] 32.0 [/math]	[math] 1.5 [/math]
[math] 9 [/math]	[math] 40.5 [/math]	[math] 4.1 [/math]
[math] 10 [/math]	[math] 50.0 [/math]	[math] 11.0 [/math]
[math] 11 [/math]	[math] 60.5 [/math]	[math] 29.9 [/math]
[math] 12 [/math]	[math] 72.0 [/math]	[math] 81.4 [/math]
[math] 13 [/math]	[math] 84.5 [/math]	[math] 221.2 [/math]

[math]g(x) > f(x)[/math] for [math]12 <= x <= 20[/math]. [math]f(x)[/math] is quadratic and has a minimum near [math]x =16[/math], as is indicated by the values in the table. After this point, [math]f(x)[/math] will start growing more rapidly than [math]g(x)[/math] and become greater than [math]g(x)[/math] for [math]x>20[/math].
[math]g(x) > f(x)[/math] for [math]x >= 12[/math]. Since exponential functions and quadratic functions can only intersect at one point, the intersection point near [math]x=11.5[/math] means that [math] g(x) > f(x)[/math] for [math]x >= 12[/math] and [math]f(x) > g(x)[/math] for [math]x <= 11[/math].
[math]g(x) > f(x)[/math] for all values of [math]x[/math]. Since [math]g(x)[/math] is exponential, it must therefore always be greater than a quadratic function.
[math]g(x) > f(x)[/math] for [math]x>=12[/math]. Since [math]g(x)[/math] is exponential, the difference of values between each unit interval will continue to increase. Although the difference of quadratic values will also increase, they do so at a much slower rate.