Mathematics Test 1 Sat Examination
Question 31
A proposal for a new library was included on an election ballot. A radio show stated that 3 times as many people voted in favor of the proposal as people who voted against it. A social media post reported that 15,000 more people voted in favor of the proposal than voted against it.
Based on these data, how many people voted against the proposal?
Let x be the number of people who voted against the proposal.
Then, according to the radio show:
Votes in favor = 3x
According to the social media post:\[3x - x = 15,000 \Rightarrow 2x = 15,000 \Rightarrow x = 7,500\]
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Question 32
Note: Figure not drawn to scale.
In the figure, lines m and n are parallel. If x = 6k + 13 and y = 8k − 29, what is the value of z ?
Because m and n are parallel, and line t cuts through them, then x = y
6k + 13 = 8k - 29
(6 - 8)k = -29 - 13
-2k = -42
k = 21
so y = 8(21) - 29 = 168 - 29 = 139
z = 180 - 139 = 41o
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Question 33
−3x + 21px = 84
In the given equation, p is a constant. The equation has no solution.
What is the value of p ?
So: \[ -3 + 21p = 0 \Rightarrow 21p = 3 \Rightarrow p = \dfrac{1}{7} \] Substitute \( p = \dfrac{1}{7} \) back into the equation: \[ x(0) = 84 \Rightarrow 0 = 84 \] This is a contradiction, so the equation has no solution when: \[ \boxed{p = \dfrac{1}{7}} \]
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Question 34
f (x) = (x − 10) (x + 13)
The function f is defined by the given equation. For what value of x does f (x) reach its minimum?
\[ \text{Given: } f(x) = (x - 10)(x + 13)\]
\[\text{Expand the expression:}\]\[ f(x) = x^2 + 13x - 10x - 130 = x^2 + 3x - 130\]
\[\text{This is a quadratic function of the form }
f(x) = ax^2 + bx + c, \text{ where } a = 1,\ b = 3\]
\[\text{The minimum occurs at:}\]\[ x = \dfrac{-b}{2a} = \dfrac{-3}{2(1)} = -\dfrac{3}{2}\]
\[ \boxed{-\dfrac{3}{2}}\]
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Question 35
The function f\[f(x)=\dfrac{1}{9}{{(x-7)}^{2}}\] gives a metal ball’s height above the ground f(x), in inches,
x seconds after it started moving on a track, where 0 ≤ x ≤ 10. Which of the following is the best interpretation of the vertex of the graph of
y = f (x) in the xy-plane?
\[\text{Given: } f(x) = \dfrac{1}{9}(x - 7)^2\]\[\text{This is a quadratic function in vertex form: } f(x) = a(x - h)^2 + k\]\[\text{Comparing, we have } a = \dfrac{1}{9},\ h = 7,\ k = 0\]\[\text{So, the vertex is at } (7, 0)\]\[\text{Since } a > 0,\ \text{the parabola opens upward, meaning the vertex is a minimum point.}\]
But from the given options, The vertex represents the moment the ball was at its lowest height.
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Question 36
If a new graph of three linear equations is created using the system of equations shown and the equation x + 4y = −16, how many solutions (x, y) will the resulting system of three equations have?
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Question 37
f (x) = 5,470(0.64)x/12 The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p% of its value the preceding year, what is the value of p ?
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