Motorists often engage in secondary tasks unrelated to driving that increase cognitive workload, resulting in fatal crashes and injuries. An International Standards Organization method for measuring a driver's cognitive workload, the detection response task (DRT), correlates well with driving outcomes, but investigation of its putative theoretical basis in terms of finite attention capacity remains limited. We address this knowledge gap using evidence-accumulation modeling of simple and choice versions of the DRT in a driving scenario. Our experiments demonstrate how dual-task load affects the parameters of evidence-accumulation models. We found that the cognitive workload induced by a secondary task (counting backward by 3s) reduced the rate of evidence accumulation, consistent with rates being sensitive to limited-capacity attention. We also found a compensatory increase in the amount of evidence required for a response and a small speeding in the time for nondecision processes. The International Standards Organization version of the DRT was found to be most sensitive to cognitive workload. A Wald-distributed evidence-accumulation model augmented with a parameter measuring response omissions provided a parsimonious measure of the underlying causes of cognitive workload in this task. This work demonstrates that evidence-accumulation modeling can accurately represent data produced by cognitive workload measurements, reproduce the data through simulation, and provide supporting evidence for the cognitive processes underlying cognitive workload. Our results provide converging evidence that the DRT method is sensitive to dynamic fluctuations in limited-capacity attention. Public Significance Statement People around the world endanger the lives of themselves and others every day by dividing their attention across multiple tasks, such as driving and talking on a cell phone. These dangers result from splitting and overtaxing our limited voluntary attentional efforts. Current tools for measuring attentional effort, also known as cognitive workload, lack insight into cognitive factors that can cause fatal errors. With the advent of new distracting technology in cars, if we do not effectively measure cognitive workload fatal human errors may grow. To quantify cognitive workload under a simulated driving-like task, the current study details our application of mathematical modeling to an International Standard for measuring ongoing cognitive workload in the vehicle. This research provides a framework for accurately quantifying cognitive workload and the factors that contribute to it, which will allow future researchers and policymakers to determine the danger inherent in many tasks within the vehicle.